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<h2 class="hd hd-2 unit-title">Introduction</h2>
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<p>Read <a href="/assets/courseware/v1/e44fc81d885fd0155393c504d2e86145/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_Session12.pdf" target="[object Object]">Chapter 8.1–8.5 (PDF)</a> of <em>Mathematics for Computer Science</em> for 2.1 GCDs.</p>
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<h2 class="hd hd-2 unit-title">Lecture Video | GCDs & Linear Combinations</h2>
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<h2 class="hd hd-2 unit-title">Lecture Video | Euclidean Algorithm</h2>
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<mi>G</mi>
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<h2 class="hd hd-2 unit-title">Exercise | GCDs 1</h2>
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GCDs I
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<math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>a</mi>
<mo>=</mo>
<mn>21212121</mn>
</math>
<br/>
<br/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>b</mi>
<mo>=</mo>
<mn>12121212</mn>
</math>
<br/>
<br/>
1. What is the GCD of <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>a</mi>
</math> and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>b</mi>
</math>?</label>
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Question 2
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<label class="problem-group-label" for="input_6d61587dda564ad3abb843af7cf10bbc_2_1" id="label_6d61587dda564ad3abb843af7cf10bbc_2_1">2. How many steps of the Euclidean algorithm are needed to compute this GCD?
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<br/>
<em>A step of the Euclidean algorithm is defined as an application of the equation: </em>
<br/>
<br/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>G</mi>
<mi>C</mi>
<mi>D</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>G</mi>
<mi>C</mi>
<mi>D</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo>,</mo>
<mi>r</mi>
<mi>e</mi>
<mi>m</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>.</mo>
</math>
<br/>
<br/>
The algorithm begins with <math xmlns="http://www.w3.org/1998/Math/MathML"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </math> and ends with <math xmlns="http://www.w3.org/1998/Math/MathML"> <mo stretchy="false">(</mo> <mi>g</mi> <mi>c</mi> <mi>d</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </math></label>
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<h2 class="hd hd-2 unit-title">Lecture Video | Revisiting Die Hard</h2>
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<h2 class="hd hd-2 unit-title">Lecture Video | Prime Factorization</h2>
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<h2 class="hd hd-2 unit-title">Exercise | Unique Primes</h2>
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Unique Primes
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<mn>40500</mn>
</math> have?</label>
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<h2 class="hd hd-2 unit-title">Exercise | Divisors</h2>
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Divisors
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<h2 class="hd hd-2 unit-title">Exercise | GCDs II</h2>
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GCDs II
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<mi>x</mi>
<mo>=</mo>
<msup>
<mn>17</mn>
<mn>88</mn>
</msup>
<mo>&#8901;<!-- &#8901; --></mo>
<msup>
<mn>31</mn>
<mn>5</mn>
</msup>
<mo>&#8901;<!-- &#8901; --></mo>
<msup>
<mn>37</mn>
<mn>2</mn>
</msup>
<mo>&#8901;<!-- &#8901; --></mo>
<msup>
<mn>59</mn>
<mn>1000</mn>
</msup>
<mspace linebreak="newline"/>
<mi>y</mi>
<mo>=</mo>
<msup>
<mn>19</mn>
<msup>
<mn>9</mn>
<mn>22</mn>
</msup>
</msup>
<mo>&#8901;<!-- &#8901; --></mo>
<msup>
<mn>37</mn>
<mn>12</mn>
</msup>
<mo>&#8901;<!-- &#8901; --></mo>
<msup>
<mn>53</mn>
<mn>3678</mn>
</msup>
<mo>&#8901;<!-- &#8901; --></mo>
<msup>
<mn>59</mn>
<mn>29</mn>
</msup>
</math></div></label>
<p class="question-description" id="description_dec5048211ec49448bba7ed46a77ba4c_1_1"> Use <math xmlns="http://www.w3.org/1998/Math/MathML">
<merror>
<mtext>^</mtext>
</merror>
</math> for exponentiation and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&#8727;<!-- &#8727; --></mo>
</math> for multiplication in your answer. </p>
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