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<h2 class="hd hd-2 unit-title">Introduction</h2>
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<p>Read <a href="/assets/courseware/v1/08cf6557ab2f20085b7171e16bd7e26a/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_Session13.pdf" target="[object Object]">Chapter 8.6–8.9 (PDF)</a> of <em>Mathematics for Computer Science</em> for 2.2 Congruences.</p>
<p>View the <a href="/assets/courseware/v1/0a36540419b4da3d3ef83fda02d35fe1/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_cp13.pdf" target="[object Object]">Section 2.2 In-Class Questions (PDF)</a></p>
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<h2 class="hd hd-2 unit-title">Lecture Video | Congruence mod n</h2>
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<p>Download a copy of the slides for <a href="/assets/courseware/v1/0906f31195856211709a7fe9d36cfc64/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_congruence.pdf" target="[object Object]">Congruence mod n (PDF)</a></p>
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<h2 class="hd hd-2 unit-title">Exercise | Divisibility and Congruence</h2>
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Divisibility and Congruence
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<h2 class="hd hd-2 unit-title">Exercise | Inverses mod n:</h2>
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Inverses mod n:
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<label class="problem-group-label" for="input_df6c2a09f5ab4de1a585900186a7cad0_2_1" id="label_df6c2a09f5ab4de1a585900186a7cad0_2_1">1. Why cannot we cancel the <math xmlns="http://www.w3.org/1998/Math/MathML">
<mn>2</mn>
</math>'s in the equation <math xmlns="http://www.w3.org/1998/Math/MathML">
<mn>8</mn>
<mo>&#8901;<!-- &#8901; --></mo>
<mn>2</mn>
<mo>&#8801;<!-- &#8801; --></mo>
<mn>3</mn>
<mo>&#8901;<!-- &#8901; --></mo>
<mn>2</mn>
<mo stretchy="false">(</mo>
<mtext>mod&#160;</mtext>
<mn>10</mn>
<mo stretchy="false">)</mo>
</math> to <math xmlns="http://www.w3.org/1998/Math/MathML">
<mn>8</mn>
<mo>&#8801;<!-- &#8801; --></mo>
<mn>3</mn>
<mo stretchy="false">(</mo>
<mtext>mod&#160;</mtext>
<mn>10</mn>
<mo stretchy="false">)</mo>
</math>?</label>
<select name="input_df6c2a09f5ab4de1a585900186a7cad0_2_1" id="input_df6c2a09f5ab4de1a585900186a7cad0_2_1" aria-describedby="status_df6c2a09f5ab4de1a585900186a7cad0_2_1">
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<option value="2 is a prime number"> 2 is a prime number</option>
<option value="2 is even"> 2 is even</option>
<option value="2 has common factors with 10"> 2 has common factors with 10</option>
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<option value="We CAN cancel the 2"> We CAN cancel the 2</option>
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<mi>k</mi>
</math> and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>n</mi>
</math> are _____, then <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>k</mi>
</math> has a multiplicative inverse in the equation <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>a</mi>
<mo>&#8901;<!-- &#8901; --></mo>
<mi>k</mi>
<mo>&#8801;<!-- &#8801; --></mo>
<mi>b</mi>
<mo>&#8901;<!-- &#8901; --></mo>
<mi>k</mi>
<mo stretchy="false">(</mo>
<mtext>mod&#160;</mtext>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math>.</label>
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<option value="collinear"> collinear</option>
<option value="of different parity"> of different parity</option>
<option value="relatively prime"> relatively prime</option>
<option value="invertible"> invertible</option>
<option value="irreducible"> irreducible</option>
<option value="distinct"> distinct</option>
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<h2 class="hd hd-2 unit-title">Exercise | Multiplicative Inverses</h2>
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Multiplicative Inverses
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<div class="unanswered ">
<label class="problem-group-label" for="input_02666a7d12b44ef78094f3eee5f70a75_2_1" id="label_02666a7d12b44ef78094f3eee5f70a75_2_1">What is the multiplicative inverse <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo stretchy="false">(</mo>
<mtext>mod&#160;</mtext>
<mn>7)</mn>
</math> of <math xmlns="http://www.w3.org/1998/Math/MathML">
<mn>2</mn>
</math> ?</label>
<p class="question-description" id="description_02666a7d12b44ef78094f3eee5f70a75_1_1"> <em>Reminder</em>: By definition, your answer must be an integer between <math xmlns="http://www.w3.org/1998/Math/MathML">
<mn>0</mn>
</math> and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mn>6</mn>
</math>.</p>
<input type="text" name="input_02666a7d12b44ef78094f3eee5f70a75_2_1" id="input_02666a7d12b44ef78094f3eee5f70a75_2_1" aria-describedby="status_02666a7d12b44ef78094f3eee5f70a75_2_1 description_02666a7d12b44ef78094f3eee5f70a75_1_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_02666a7d12b44ef78094f3eee5f70a75_2_1"/>
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<h2 class="hd hd-2 unit-title">Exercise | Inverses With Linear Combinations</h2>
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Inverses With Linear Combinations
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_e8936fcc58f347b0a0b6aff2ca47cac1_2_1" class=" capa_inputtype textline">
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<label class="problem-group-label" for="input_e8936fcc58f347b0a0b6aff2ca47cac1_2_1" id="label_e8936fcc58f347b0a0b6aff2ca47cac1_2_1">What is the inverse <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo stretchy="false">(</mo>
<mtext>mod&#160;</mtext>
<mn>25</mn>
<mo stretchy="false">)</mo>
</math> of <math xmlns="http://www.w3.org/1998/Math/MathML">
<mn>32</mn>
</math> ?</label>
<p class="question-description" id="description_e8936fcc58f347b0a0b6aff2ca47cac1_1_1"> Reminder: Your answer must be an integer in the interval <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo stretchy="false">[</mo>
<mn>0</mn>
<mo>,</mo>
<mn>25</mn>
<mo stretchy="false">)</mo>
</math> </p>
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