<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="3c10df70e12211ef932016ffeba94eb7" data-block-type="vertical" data-graded="False" data-runtime-version="1" data-init="VerticalStudentView" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@d525acbe90804359adebb6e927c1f595">
<h2 class="hd hd-2 unit-title">Introduction</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@html+block@e32539bf1a5b4fcd8e06cbf312e8e98e">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="3c10df70e12211ef932016ffeba94eb7" data-block-type="html" data-graded="False" data-runtime-version="1" data-init="XBlockToXModuleShim" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@html+block@e32539bf1a5b4fcd8e06cbf312e8e98e">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>Read <a href="/assets/courseware/v1/22d8346e384aca3ccd44033a6fbe1b2f/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_Session14.pdf" target="[object Object]">Chapter 8.10 (PDF)</a> of <em>Mathematics for Computer Science</em> for 2.3 Euler's Theorem.</p>
<p>View the <a href="/assets/courseware/v1/41dfcedaf37dca34247a5e5b1c81fab2/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_cp14.pdf" target="[object Object]">Section 2.3 In-Class Questions (PDF)</a></p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="3c10df70e12211ef932016ffeba94eb7" data-block-type="vertical" data-graded="False" data-runtime-version="1" data-init="VerticalStudentView" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@b6f428dcf95446f0b2d766cc69fc9650">
<h2 class="hd hd-2 unit-title">Lecture Video | Modular Exponentiation: Euler's Function</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@video+block@8b212c79ade44d5684e0369987e6b062">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="3c10df70e12211ef932016ffeba94eb7" data-block-type="video" data-graded="False" data-runtime-version="1" data-init="XBlockToXModuleShim" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@video+block@8b212c79ade44d5684e0369987e6b062">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Modular Exponentiation: Euler's Function</h3>
<div
id="video_8b212c79ade44d5684e0369987e6b062"
class="video closed"
data-metadata='{"streams": "1.00:TeRYL7kkhqs", "autohideHtml5": false, "autoplay": false, "transcriptTranslationUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@8b212c79ade44d5684e0369987e6b062/handler/transcript/translation/__lang__", "transcriptLanguage": "en", "publishCompletionUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@8b212c79ade44d5684e0369987e6b062/handler/publish_completion", "transcriptAvailableTranslationsUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@8b212c79ade44d5684e0369987e6b062/handler/transcript/available_translations", "completionEnabled": false, "generalSpeed": 1.0, "ytTestTimeout": 1500, "ytMetadataEndpoint": "", "ytApiUrl": "https://www.youtube.com/iframe_api", "showCaptions": "true", "recordedYoutubeIsAvailable": true, "sources": ["https://ia800207.us.archive.org/32/items/MIT6.042JS15/MIT6_042JS15_phi_ipod.mp4"], "end": 0.0, "saveStateEnabled": false, "prioritizeHls": false, "savedVideoPosition": 0.0, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "start": 0.0, "saveStateUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@8b212c79ade44d5684e0369987e6b062/handler/xmodule_handler/save_user_state", "speed": null, "autoAdvance": false, "completionPercentage": 0.95, "duration": 0.0, "lmsRootURL": "https://openlearninglibrary.mit.edu", "poster": null}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="8b212c79ade44d5684e0369987e6b062"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_8b212c79ade44d5684e0369987e6b062">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_8b212c79ade44d5684e0369987e6b062">
<div class="wrapper-download-video">
<h4 class="hd hd-5">Video</h4>
<a class="btn-link video-sources video-download-button" href="https://ia800207.us.archive.org/32/items/MIT6.042JS15/MIT6_042JS15_phi_ipod.mp4">
Download video file
</a>
</div>
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@8b212c79ade44d5684e0369987e6b062/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@8b212c79ade44d5684e0369987e6b062/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:OCW+6.042J+2T2019+type@html+block@af3711caa96643df9c88ebf68049434c">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="3c10df70e12211ef932016ffeba94eb7" data-block-type="html" data-graded="False" data-runtime-version="1" data-init="XBlockToXModuleShim" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@html+block@af3711caa96643df9c88ebf68049434c">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>Download a copy of the slides for <a href="/assets/courseware/v1/b954cf479bdbfb07757a70f8d5c264a5/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_ModularEuler.pdf" target="[object Object]">Modular Exponentiation: Euler's Function (PDF)</a></p>
<p><a href="/assets/courseware/v1/01477d5cdc61072976219afc1bfd6b5e/asset-v1:OCW+6.042J+2T2019+type@asset+block/Modular_Ex_2.3_Lectrans.pdf" target="[object Object]">Lecture video transcript (PDF)</a></p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="3c10df70e12211ef932016ffeba94eb7" data-block-type="vertical" data-graded="False" data-runtime-version="1" data-init="VerticalStudentView" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@9a73900b9efe4ed9be4b55174f1f8ca0">
<h2 class="hd hd-2 unit-title">Exercise | Euler's Totient Function</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@problem+block@0e4bc115ee7e4c4a9f12f935431d5547">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="3c10df70e12211ef932016ffeba94eb7" data-block-type="problem" data-graded="False" data-runtime-version="1" data-init="XBlockToXModuleShim" data-has-score="True" data-usage-id="block-v1:OCW+6.042J+2T2019+type@problem+block@0e4bc115ee7e4c4a9f12f935431d5547">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_0e4bc115ee7e4c4a9f12f935431d5547" class="problems-wrapper" role="group"
aria-labelledby="0e4bc115ee7e4c4a9f12f935431d5547-problem-title"
data-problem-id="block-v1:OCW+6.042J+2T2019+type@problem+block@0e4bc115ee7e4c4a9f12f935431d5547" data-url="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@problem+block@0e4bc115ee7e4c4a9f12f935431d5547/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="0e4bc115ee7e4c4a9f12f935431d5547-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@0e4bc115ee7e4c4a9f12f935431d5547-problem-progress" tabindex="-1">
Euler&#39;s Totient Function
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@0e4bc115ee7e4c4a9f12f935431d5547-problem-progress"></div>
<div class="problem">
<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_0e4bc115ee7e4c4a9f12f935431d5547_2_1">
<fieldset aria-describedby="status_0e4bc115ee7e4c4a9f12f935431d5547_2_1">
<legend id="0e4bc115ee7e4c4a9f12f935431d5547_2_1-legend" class="response-fieldset-legend field-group-hd">Which of the following are equal to <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi mathvariant="normal">&#934;<!-- &#934; --></mi>
<mo stretchy="false">(</mo>
<mn>300</mn>
<mo stretchy="false">)</mo>
</math>?</legend>
<div class="field">
<input type="checkbox" name="input_0e4bc115ee7e4c4a9f12f935431d5547_2_1[]" id="input_0e4bc115ee7e4c4a9f12f935431d5547_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="0e4bc115ee7e4c4a9f12f935431d5547_2_1-choice_0-label" for="input_0e4bc115ee7e4c4a9f12f935431d5547_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_0e4bc115ee7e4c4a9f12f935431d5547_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi mathvariant="normal">&#934;<!-- &#934; --></mi>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>&#8901;<!-- &#8901; --></mo>
<mi mathvariant="normal">&#934;<!-- &#934; --></mi>
<mo stretchy="false">(</mo>
<mn>100</mn>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_0e4bc115ee7e4c4a9f12f935431d5547_2_1[]" id="input_0e4bc115ee7e4c4a9f12f935431d5547_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="0e4bc115ee7e4c4a9f12f935431d5547_2_1-choice_1-label" for="input_0e4bc115ee7e4c4a9f12f935431d5547_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_0e4bc115ee7e4c4a9f12f935431d5547_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi mathvariant="normal">&#934;<!-- &#934; --></mi>
<mo stretchy="false">(</mo>
<mn>10</mn>
<mo stretchy="false">)</mo>
<mo>&#8901;<!-- &#8901; --></mo>
<mi mathvariant="normal">&#934;<!-- &#934; --></mi>
<mo stretchy="false">(</mo>
<mn>30</mn>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_0e4bc115ee7e4c4a9f12f935431d5547_2_1[]" id="input_0e4bc115ee7e4c4a9f12f935431d5547_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="0e4bc115ee7e4c4a9f12f935431d5547_2_1-choice_2-label" for="input_0e4bc115ee7e4c4a9f12f935431d5547_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_0e4bc115ee7e4c4a9f12f935431d5547_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi mathvariant="normal">&#934;<!-- &#934; --></mi>
<mo stretchy="false">(</mo>
<mn>2</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
<mo>&#8901;<!-- &#8901; --></mo>
<mi mathvariant="normal">&#934;<!-- &#934; --></mi>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>&#8901;<!-- &#8901; --></mo>
<mi mathvariant="normal">&#934;<!-- &#934; --></mi>
<mo stretchy="false">(</mo>
<mn>25</mn>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_0e4bc115ee7e4c4a9f12f935431d5547_2_1[]" id="input_0e4bc115ee7e4c4a9f12f935431d5547_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="0e4bc115ee7e4c4a9f12f935431d5547_2_1-choice_3-label" for="input_0e4bc115ee7e4c4a9f12f935431d5547_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_0e4bc115ee7e4c4a9f12f935431d5547_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi mathvariant="normal">&#934;<!-- &#934; --></mi>
<mo stretchy="false">(</mo>
<mn>4</mn>
<mo stretchy="false">)</mo>
<mo>&#8901;<!-- &#8901; --></mo>
<mi mathvariant="normal">&#934;<!-- &#934; --></mi>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>&#8901;<!-- &#8901; --></mo>
<mi mathvariant="normal">&#934;<!-- &#934; --></mi>
<mo stretchy="false">(</mo>
<mn>25</mn>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_0e4bc115ee7e4c4a9f12f935431d5547_2_1[]" id="input_0e4bc115ee7e4c4a9f12f935431d5547_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="0e4bc115ee7e4c4a9f12f935431d5547_2_1-choice_4-label" for="input_0e4bc115ee7e4c4a9f12f935431d5547_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_0e4bc115ee7e4c4a9f12f935431d5547_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi mathvariant="normal">&#934;<!-- &#934; --></mi>
<mo stretchy="false">(</mo>
<mn>150</mn>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<span id="answer_0e4bc115ee7e4c4a9f12f935431d5547_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_0e4bc115ee7e4c4a9f12f935431d5547_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_0e4bc115ee7e4c4a9f12f935431d5547_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Euler&#39;s Totient Function " />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_0e4bc115ee7e4c4a9f12f935431d5547" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_0e4bc115ee7e4c4a9f12f935431d5547">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="reset problem-action-btn btn-default btn-small" data-value="Reset"><span class="icon fa fa-refresh" aria-hidden="true"></span><span aria-hidden="true">Reset</span><span class="sr">Reset your answer</span></button>
</span>
<span class="problem-action-button-wrapper">
<button type="button" class="show problem-action-btn btn-default btn-small" aria-describedby="0e4bc115ee7e4c4a9f12f935431d5547-problem-title"><span class="icon fa fa-info-circle" aria-hidden="true"></span><span class="show-label">Show Answer</span></button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="0e4bc115ee7e4c4a9f12f935431d5547-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="0e4bc115ee7e4c4a9f12f935431d5547-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="0e4bc115ee7e4c4a9f12f935431d5547-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="3c10df70e12211ef932016ffeba94eb7" data-block-type="vertical" data-graded="False" data-runtime-version="1" data-init="VerticalStudentView" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@7c08cb5726e24dc792e317547d6190ee">
<h2 class="hd hd-2 unit-title">Lecture Video | The Ring Zn</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@video+block@432a37afef2049d59e93cc6440c09de0">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="3c10df70e12211ef932016ffeba94eb7" data-block-type="video" data-graded="False" data-runtime-version="1" data-init="XBlockToXModuleShim" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@video+block@432a37afef2049d59e93cc6440c09de0">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">The Ring Zn</h3>
<div
id="video_432a37afef2049d59e93cc6440c09de0"
class="video closed"
data-metadata='{"streams": "1.00:dZgI16nMuqE", "autohideHtml5": false, "autoplay": false, "transcriptTranslationUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@432a37afef2049d59e93cc6440c09de0/handler/transcript/translation/__lang__", "transcriptLanguage": "en", "publishCompletionUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@432a37afef2049d59e93cc6440c09de0/handler/publish_completion", "transcriptAvailableTranslationsUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@432a37afef2049d59e93cc6440c09de0/handler/transcript/available_translations", "completionEnabled": false, "generalSpeed": 1.0, "ytTestTimeout": 1500, "ytMetadataEndpoint": "", "ytApiUrl": "https://www.youtube.com/iframe_api", "showCaptions": "true", "recordedYoutubeIsAvailable": true, "sources": ["https://ia800207.us.archive.org/32/items/MIT6.042JS15/MIT6_042JS15_zmodn_ipod.mp4"], "end": 0.0, "saveStateEnabled": false, "prioritizeHls": false, "savedVideoPosition": 0.0, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "start": 0.0, "saveStateUrl": "/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@432a37afef2049d59e93cc6440c09de0/handler/xmodule_handler/save_user_state", "speed": null, "autoAdvance": false, "completionPercentage": 0.95, "duration": 0.0, "lmsRootURL": "https://openlearninglibrary.mit.edu", "poster": null}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="432a37afef2049d59e93cc6440c09de0"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_432a37afef2049d59e93cc6440c09de0">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_432a37afef2049d59e93cc6440c09de0">
<div class="wrapper-download-video">
<h4 class="hd hd-5">Video</h4>
<a class="btn-link video-sources video-download-button" href="https://ia800207.us.archive.org/32/items/MIT6.042JS15/MIT6_042JS15_zmodn_ipod.mp4">
Download video file
</a>
</div>
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@432a37afef2049d59e93cc6440c09de0/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@video+block@432a37afef2049d59e93cc6440c09de0/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:OCW+6.042J+2T2019+type@html+block@2435c674ba22425cb1343e0a9c761ef0">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="3c10df70e12211ef932016ffeba94eb7" data-block-type="html" data-graded="False" data-runtime-version="1" data-init="XBlockToXModuleShim" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@html+block@2435c674ba22425cb1343e0a9c761ef0">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>Download a copy of the slides for <a href="/assets/courseware/v1/a71a2334b0a1ccc497ee7e23c95838f8/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_TheRingZn.pdf" target="[object Object]">The Ring Z<sub>n</sub> (PDF)</a></p>
<p><a href="/assets/courseware/v1/173c08e5cfe073f8817e79709c684fbf/asset-v1:OCW+6.042J+2T2019+type@asset+block/Ringz_2.3_Lectrans.pdf" target="[object Object]">Lecture video transcript (PDF)</a></p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="3c10df70e12211ef932016ffeba94eb7" data-block-type="vertical" data-graded="False" data-runtime-version="1" data-init="VerticalStudentView" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@8e8d43c0a29049a0a50412b5b040faf6">
<h2 class="hd hd-2 unit-title">Exercise | The Ring</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@problem+block@c147d813700d443b850f93c1644605de">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="3c10df70e12211ef932016ffeba94eb7" data-block-type="problem" data-graded="False" data-runtime-version="1" data-init="XBlockToXModuleShim" data-has-score="True" data-usage-id="block-v1:OCW+6.042J+2T2019+type@problem+block@c147d813700d443b850f93c1644605de">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_c147d813700d443b850f93c1644605de" class="problems-wrapper" role="group"
aria-labelledby="c147d813700d443b850f93c1644605de-problem-title"
data-problem-id="block-v1:OCW+6.042J+2T2019+type@problem+block@c147d813700d443b850f93c1644605de" data-url="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@problem+block@c147d813700d443b850f93c1644605de/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="2"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="c147d813700d443b850f93c1644605de-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@c147d813700d443b850f93c1644605de-problem-progress" tabindex="-1">
The Ring
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@c147d813700d443b850f93c1644605de-problem-progress"></div>
<div class="problem">
<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_c147d813700d443b850f93c1644605de_2_1">
<fieldset aria-describedby="status_c147d813700d443b850f93c1644605de_2_1">
<legend id="c147d813700d443b850f93c1644605de_2_1-legend" class="response-fieldset-legend field-group-hd">1. Which of the following ALWAYS hold in <math xmlns="http://www.w3.org/1998/Math/MathML">
<msub>
<mi>Z</mi>
<mi>n</mi>
</msub>
</math>?</legend>
<div class="field">
<input type="checkbox" name="input_c147d813700d443b850f93c1644605de_2_1[]" id="input_c147d813700d443b850f93c1644605de_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="c147d813700d443b850f93c1644605de_2_1-choice_0-label" for="input_c147d813700d443b850f93c1644605de_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_c147d813700d443b850f93c1644605de_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mo>=</mo>
<mi>b</mi>
<mo>+</mo>
<mi>a</mi>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_c147d813700d443b850f93c1644605de_2_1[]" id="input_c147d813700d443b850f93c1644605de_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="c147d813700d443b850f93c1644605de_2_1-choice_1-label" for="input_c147d813700d443b850f93c1644605de_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_c147d813700d443b850f93c1644605de_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>a</mi>
<mi>b</mi>
<mo>=</mo>
<mi>b</mi>
<mi>c</mi>
<mspace width="thickmathspace"/>
<mo stretchy="false">&#10233;<!-- &#10233; --></mo>
<mspace width="thickmathspace"/>
<mi>a</mi>
<mo>=</mo>
<mi>c</mi>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_c147d813700d443b850f93c1644605de_2_1[]" id="input_c147d813700d443b850f93c1644605de_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="c147d813700d443b850f93c1644605de_2_1-choice_2-label" for="input_c147d813700d443b850f93c1644605de_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_c147d813700d443b850f93c1644605de_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo stretchy="false">(</mo>
<mi>a</mi>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mi>c</mi>
<mo>=</mo>
<mi>a</mi>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mi>c</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_c147d813700d443b850f93c1644605de_2_1[]" id="input_c147d813700d443b850f93c1644605de_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="c147d813700d443b850f93c1644605de_2_1-choice_3-label" for="input_c147d813700d443b850f93c1644605de_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_c147d813700d443b850f93c1644605de_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>a</mi>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo>+</mo>
<mi>c</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>a</mi>
<mi>b</mi>
<mo>+</mo>
<mi>a</mi>
<mi>c</mi>
</math>
</label>
</div>
<span id="answer_c147d813700d443b850f93c1644605de_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_c147d813700d443b850f93c1644605de_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_c147d813700d443b850f93c1644605de_3_1" class=" capa_inputtype textline">
<div class="unanswered ">
<label class="problem-group-label" for="input_c147d813700d443b850f93c1644605de_3_1" id="label_c147d813700d443b850f93c1644605de_3_1">2. What are the elements of <math xmlns="http://www.w3.org/1998/Math/MathML">
<msubsup>
<mi>Z</mi>
<mn>7</mn>
<mo>&#8727;<!-- &#8727; --></mo>
</msubsup>
</math>?</label>
<p class="question-description" id="description_c147d813700d443b850f93c1644605de_2_1"> Enter your answer as series of numbers in INCREASING order, separated by spaces (e.g. 2 4 6 8)
</p>
<input type="text" name="input_c147d813700d443b850f93c1644605de_3_1" id="input_c147d813700d443b850f93c1644605de_3_1" aria-describedby="status_c147d813700d443b850f93c1644605de_3_1 description_c147d813700d443b850f93c1644605de_2_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_c147d813700d443b850f93c1644605de_3_1"/>
<span class="status unanswered" id="status_c147d813700d443b850f93c1644605de_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_c147d813700d443b850f93c1644605de_3_1" class="answer"/>
</div>
</div></div>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="The Ring" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_c147d813700d443b850f93c1644605de" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_c147d813700d443b850f93c1644605de">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="reset problem-action-btn btn-default btn-small" data-value="Reset"><span class="icon fa fa-refresh" aria-hidden="true"></span><span aria-hidden="true">Reset</span><span class="sr">Reset your answer</span></button>
</span>
<span class="problem-action-button-wrapper">
<button type="button" class="show problem-action-btn btn-default btn-small" aria-describedby="c147d813700d443b850f93c1644605de-problem-title"><span class="icon fa fa-info-circle" aria-hidden="true"></span><span class="show-label">Show Answer</span></button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="c147d813700d443b850f93c1644605de-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="c147d813700d443b850f93c1644605de-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="c147d813700d443b850f93c1644605de-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="3c10df70e12211ef932016ffeba94eb7" data-block-type="vertical" data-graded="False" data-runtime-version="1" data-init="VerticalStudentView" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@f7f36cfa54804f8f961d2299a7d82727">
<h2 class="hd hd-2 unit-title">Exercise | Z mod n</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@problem+block@c6ce13fd5fc2402ba453371f388499dd">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="3c10df70e12211ef932016ffeba94eb7" data-block-type="problem" data-graded="False" data-runtime-version="1" data-init="XBlockToXModuleShim" data-has-score="True" data-usage-id="block-v1:OCW+6.042J+2T2019+type@problem+block@c6ce13fd5fc2402ba453371f388499dd">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_c6ce13fd5fc2402ba453371f388499dd" class="problems-wrapper" role="group"
aria-labelledby="c6ce13fd5fc2402ba453371f388499dd-problem-title"
data-problem-id="block-v1:OCW+6.042J+2T2019+type@problem+block@c6ce13fd5fc2402ba453371f388499dd" data-url="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@problem+block@c6ce13fd5fc2402ba453371f388499dd/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="3"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="c6ce13fd5fc2402ba453371f388499dd-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@c6ce13fd5fc2402ba453371f388499dd-problem-progress" tabindex="-1">
Z mod n
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@c6ce13fd5fc2402ba453371f388499dd-problem-progress"></div>
<div class="problem">
<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p> 1. Solve for <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML">
<ns0:mi>x</ns0:mi>
</ns0:math>:</p>
<div id="inputtype_c6ce13fd5fc2402ba453371f388499dd_2_1" class=" capa_inputtype textline">
<div class="unanswered ">
<label class="problem-group-label" for="input_c6ce13fd5fc2402ba453371f388499dd_2_1" id="label_c6ce13fd5fc2402ba453371f388499dd_2_1"><math xmlns="http://www.w3.org/1998/Math/MathML">
<mn>9</mn>
<mo>+</mo>
<mn>12</mn>
<mo>=</mo>
<mi>x</mi>
<mtext>&#160;</mtext>
<mo stretchy="false">(</mo>
<msub>
<mi mathvariant="double-struck">Z</mi>
<mn>13</mn>
</msub>
<mo stretchy="false">)</mo>
</math></label>
<input type="text" name="input_c6ce13fd5fc2402ba453371f388499dd_2_1" id="input_c6ce13fd5fc2402ba453371f388499dd_2_1" aria-describedby="status_c6ce13fd5fc2402ba453371f388499dd_2_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_c6ce13fd5fc2402ba453371f388499dd_2_1"/>
<span class="status unanswered" id="status_c6ce13fd5fc2402ba453371f388499dd_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_c6ce13fd5fc2402ba453371f388499dd_2_1" class="answer"/>
</div>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><p> 2. Solve for <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML">
<ns0:mi>x</ns0:mi>
</ns0:math>: </p>
<div id="inputtype_c6ce13fd5fc2402ba453371f388499dd_3_1" class=" capa_inputtype textline">
<div class="unanswered ">
<label class="problem-group-label" for="input_c6ce13fd5fc2402ba453371f388499dd_3_1" id="label_c6ce13fd5fc2402ba453371f388499dd_3_1"><math xmlns="http://www.w3.org/1998/Math/MathML">
<mn>7</mn>
<mo>&#8901;<!-- &#8901; --></mo>
<mn>5</mn>
<mo>=</mo>
<mi>x</mi>
<mtext>&#160;</mtext>
<mo stretchy="false">(</mo>
<msub>
<mi mathvariant="double-struck">Z</mi>
<mn>8</mn>
</msub>
<mo stretchy="false">)</mo>
</math></label>
<input type="text" name="input_c6ce13fd5fc2402ba453371f388499dd_3_1" id="input_c6ce13fd5fc2402ba453371f388499dd_3_1" aria-describedby="status_c6ce13fd5fc2402ba453371f388499dd_3_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_c6ce13fd5fc2402ba453371f388499dd_3_1"/>
<span class="status unanswered" id="status_c6ce13fd5fc2402ba453371f388499dd_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_c6ce13fd5fc2402ba453371f388499dd_3_1" class="answer"/>
</div>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="inputtype option-input ">
<label class="problem-group-label" for="input_c6ce13fd5fc2402ba453371f388499dd_4_1" id="label_c6ce13fd5fc2402ba453371f388499dd_4_1">3. What is the impact of multiplying members of <math xmlns="http://www.w3.org/1998/Math/MathML">
<msubsup>
<mi mathvariant="double-struck">Z</mi>
<mi>n</mi>
<mo>&#8727;<!-- &#8727; --></mo>
</msubsup>
</math> by another member of <math xmlns="http://www.w3.org/1998/Math/MathML">
<msubsup>
<mi mathvariant="double-struck">Z</mi>
<mi>n</mi>
<mo>&#8727;<!-- &#8727; --></mo>
</msubsup>
</math>?</label>
<select name="input_c6ce13fd5fc2402ba453371f388499dd_4_1" id="input_c6ce13fd5fc2402ba453371f388499dd_4_1" aria-describedby="status_c6ce13fd5fc2402ba453371f388499dd_4_1">
<option value="option_c6ce13fd5fc2402ba453371f388499dd_4_1_dummy_default">Select an option</option>
<option value="The set of numbers and their order always stay the same"> The set of numbers and their order always stay the same</option>
<option value="The set of numbers stays the same but the order may change"> The set of numbers stays the same but the order may change</option>
<option value="We get members of Zn - Zn*"> We get members of Zn - Zn*</option>
<option value="We cannot tell"> We cannot tell</option>
</select>
<div class="indicator-container">
<span class="status unanswered" id="status_c6ce13fd5fc2402ba453371f388499dd_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
<p class="answer" id="answer_c6ce13fd5fc2402ba453371f388499dd_4_1"/>
</div></div>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Z mod n" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_c6ce13fd5fc2402ba453371f388499dd" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_c6ce13fd5fc2402ba453371f388499dd">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="reset problem-action-btn btn-default btn-small" data-value="Reset"><span class="icon fa fa-refresh" aria-hidden="true"></span><span aria-hidden="true">Reset</span><span class="sr">Reset your answer</span></button>
</span>
<span class="problem-action-button-wrapper">
<button type="button" class="show problem-action-btn btn-default btn-small" aria-describedby="c6ce13fd5fc2402ba453371f388499dd-problem-title"><span class="icon fa fa-info-circle" aria-hidden="true"></span><span class="show-label">Show Answer</span></button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="c6ce13fd5fc2402ba453371f388499dd-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="c6ce13fd5fc2402ba453371f388499dd-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="c6ce13fd5fc2402ba453371f388499dd-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="3c10df70e12211ef932016ffeba94eb7" data-block-type="vertical" data-graded="False" data-runtime-version="1" data-init="VerticalStudentView" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@9daa802a598c41d79a4868f794d7b542">
<h2 class="hd hd-2 unit-title">Exercise | Fermat's Little Theorem</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@problem+block@496f35bc5db34c669fbd3a91409caf0a">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="3c10df70e12211ef932016ffeba94eb7" data-block-type="problem" data-graded="False" data-runtime-version="1" data-init="XBlockToXModuleShim" data-has-score="True" data-usage-id="block-v1:OCW+6.042J+2T2019+type@problem+block@496f35bc5db34c669fbd3a91409caf0a">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_496f35bc5db34c669fbd3a91409caf0a" class="problems-wrapper" role="group"
aria-labelledby="496f35bc5db34c669fbd3a91409caf0a-problem-title"
data-problem-id="block-v1:OCW+6.042J+2T2019+type@problem+block@496f35bc5db34c669fbd3a91409caf0a" data-url="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@problem+block@496f35bc5db34c669fbd3a91409caf0a/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="496f35bc5db34c669fbd3a91409caf0a-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@496f35bc5db34c669fbd3a91409caf0a-problem-progress" tabindex="-1">
Fermat&#39;s Little Theorem
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@496f35bc5db34c669fbd3a91409caf0a-problem-progress"></div>
<div class="problem">
<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_496f35bc5db34c669fbd3a91409caf0a_2_1" class=" capa_inputtype textline">
<div class="unanswered ">
<label class="problem-group-label" for="input_496f35bc5db34c669fbd3a91409caf0a_2_1" id="label_496f35bc5db34c669fbd3a91409caf0a_2_1">What is <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>r</mi>
<mi>e</mi>
<mi>m</mi>
<mo stretchy="false">(</mo>
<msup>
<mn>24</mn>
<mn>78</mn>
</msup>
<mo>,</mo>
<mn>79</mn>
<mo stretchy="false">)</mo>
</math>?</label>
<p class="question-description" id="description_496f35bc5db34c669fbd3a91409caf0a_1_1"><em>Hint:</em> You should not need to do any calculation!</p>
<input type="text" name="input_496f35bc5db34c669fbd3a91409caf0a_2_1" id="input_496f35bc5db34c669fbd3a91409caf0a_2_1" aria-describedby="status_496f35bc5db34c669fbd3a91409caf0a_2_1 description_496f35bc5db34c669fbd3a91409caf0a_1_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_496f35bc5db34c669fbd3a91409caf0a_2_1"/>
<span class="status unanswered" id="status_496f35bc5db34c669fbd3a91409caf0a_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_496f35bc5db34c669fbd3a91409caf0a_2_1" class="answer"/>
</div>
</div></div>
<div class="solution-span">
<span id="solution_496f35bc5db34c669fbd3a91409caf0a_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Fermat&#39;s Little Theorem" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_496f35bc5db34c669fbd3a91409caf0a" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_496f35bc5db34c669fbd3a91409caf0a">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="reset problem-action-btn btn-default btn-small" data-value="Reset"><span class="icon fa fa-refresh" aria-hidden="true"></span><span aria-hidden="true">Reset</span><span class="sr">Reset your answer</span></button>
</span>
<span class="problem-action-button-wrapper">
<button type="button" class="show problem-action-btn btn-default btn-small" aria-describedby="496f35bc5db34c669fbd3a91409caf0a-problem-title"><span class="icon fa fa-info-circle" aria-hidden="true"></span><span class="show-label">Show Answer</span></button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="496f35bc5db34c669fbd3a91409caf0a-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="496f35bc5db34c669fbd3a91409caf0a-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="496f35bc5db34c669fbd3a91409caf0a-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="3c10df70e12211ef932016ffeba94eb7" data-block-type="vertical" data-graded="False" data-runtime-version="1" data-init="VerticalStudentView" data-has-score="False" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@122867be4f524d44b0ff598ea77a938d">
<h2 class="hd hd-2 unit-title">Exercise | Euler's Theorem</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@problem+block@27ae71cb0dc049c4bc96269eea1e1da2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:OCW+6.042J+2T2019" data-runtime-class="LmsRuntime" data-request-token="3c10df70e12211ef932016ffeba94eb7" data-block-type="problem" data-graded="False" data-runtime-version="1" data-init="XBlockToXModuleShim" data-has-score="True" data-usage-id="block-v1:OCW+6.042J+2T2019+type@problem+block@27ae71cb0dc049c4bc96269eea1e1da2">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_27ae71cb0dc049c4bc96269eea1e1da2" class="problems-wrapper" role="group"
aria-labelledby="27ae71cb0dc049c4bc96269eea1e1da2-problem-title"
data-problem-id="block-v1:OCW+6.042J+2T2019+type@problem+block@27ae71cb0dc049c4bc96269eea1e1da2" data-url="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@problem+block@27ae71cb0dc049c4bc96269eea1e1da2/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="2"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="27ae71cb0dc049c4bc96269eea1e1da2-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@27ae71cb0dc049c4bc96269eea1e1da2-problem-progress" tabindex="-1">
Euler&#39;s Theorem
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@27ae71cb0dc049c4bc96269eea1e1da2-problem-progress"></div>
<div class="problem">
<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_27ae71cb0dc049c4bc96269eea1e1da2_2_1" class=" capa_inputtype textline">
<div class="unanswered ">
<label class="problem-group-label" for="input_27ae71cb0dc049c4bc96269eea1e1da2_2_1" id="label_27ae71cb0dc049c4bc96269eea1e1da2_2_1">1. What is the value of <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi mathvariant="normal">&#934;<!-- &#934; --></mi>
<mo stretchy="false">(</mo>
<mn>175</mn>
<mo stretchy="false">)</mo>
</math>, where <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi mathvariant="normal">&#934;<!-- &#934; --></mi>
</math> is Euler's Totient function?</label>
<input type="text" name="input_27ae71cb0dc049c4bc96269eea1e1da2_2_1" id="input_27ae71cb0dc049c4bc96269eea1e1da2_2_1" aria-describedby="status_27ae71cb0dc049c4bc96269eea1e1da2_2_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_27ae71cb0dc049c4bc96269eea1e1da2_2_1"/>
<span class="status unanswered" id="status_27ae71cb0dc049c4bc96269eea1e1da2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_27ae71cb0dc049c4bc96269eea1e1da2_2_1" class="answer"/>
</div>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_27ae71cb0dc049c4bc96269eea1e1da2_3_1" class=" capa_inputtype textline">
<div class="unanswered ">
<label class="problem-group-label" for="input_27ae71cb0dc049c4bc96269eea1e1da2_3_1" id="label_27ae71cb0dc049c4bc96269eea1e1da2_3_1">2. What is the remainder of <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mn>22</mn>
<mn>12001</mn>
</msup>
</math> divided by 175?</label>
<input type="text" name="input_27ae71cb0dc049c4bc96269eea1e1da2_3_1" id="input_27ae71cb0dc049c4bc96269eea1e1da2_3_1" aria-describedby="status_27ae71cb0dc049c4bc96269eea1e1da2_3_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_27ae71cb0dc049c4bc96269eea1e1da2_3_1"/>
<span class="status unanswered" id="status_27ae71cb0dc049c4bc96269eea1e1da2_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_27ae71cb0dc049c4bc96269eea1e1da2_3_1" class="answer"/>
</div>
</div></div>
<div class="solution-span">
<span id="solution_27ae71cb0dc049c4bc96269eea1e1da2_solution_1"/>
</div><div class="solution-span">
<span id="solution_27ae71cb0dc049c4bc96269eea1e1da2_solution_2"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Euler&#39;s Theorem" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_27ae71cb0dc049c4bc96269eea1e1da2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_27ae71cb0dc049c4bc96269eea1e1da2">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="reset problem-action-btn btn-default btn-small" data-value="Reset"><span class="icon fa fa-refresh" aria-hidden="true"></span><span aria-hidden="true">Reset</span><span class="sr">Reset your answer</span></button>
</span>
<span class="problem-action-button-wrapper">
<button type="button" class="show problem-action-btn btn-default btn-small" aria-describedby="27ae71cb0dc049c4bc96269eea1e1da2-problem-title"><span class="icon fa fa-info-circle" aria-hidden="true"></span><span class="show-label">Show Answer</span></button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="27ae71cb0dc049c4bc96269eea1e1da2-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="27ae71cb0dc049c4bc96269eea1e1da2-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="27ae71cb0dc049c4bc96269eea1e1da2-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>