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326 lines
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Greatest Common Factor</h1>
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<p class="center"><i>The highest number that divides exactly into two or more numbers.<br>
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It is the "greatest" thing for simplifying fractions!</i></p>
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<h3>Let's start with an Example ... </h3>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="numbers/images/greatest-common-factor.svg" style="background-color:white; border:5px solid white; border-radius:5px;" alt="greatest common factor"></p>
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<h3>Greatest Common Factor of 12 and 16</h3>
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<ol>
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<li>Find all the <b>Factors</b> of each number,</li>
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<li>Circle the <b>Common</b> factors,</li>
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<li>Choose the <b>Greatest</b> of those</li>
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</ol>
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<div style="clear:both"></div>
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</div>
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<h2>So ... what is a "Factor" ?</h2>
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<p>Factors are numbers we can multiply together to get another number:</p>
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<div class="center"><img src="numbers/images/factor-2x3.svg" alt="factors"></div>
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<p>A number can have many factors:</p>
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<div class="example">
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<p>Factors of 12 are <b>1, 2, 3, 4, 6</b> and <b>12 </b>...<br>
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<br>
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...
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because <b>2</b> × <b>6</b> = 12, or <b>4</b> × <b>3</b> = 12, or <b>1</b> × <b>12</b> = 12.</p>
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</div>
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<p>(Read how to find <a href="numbers/factors-all-tool.html">All the Factors of a Number</a>. In our case we don't need the negative ones.)</p>
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<h2>What is a "Common Factor" ?</h2>
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<p>Say we have worked out the factors of two numbers:</p>
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<div class="example">
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<h3>Example: Factors of 12 and 30</h3>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td>Factors of 12 are <b>1, 2, 3, 4, 6 </b>and<b> 12</b></td>
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</tr>
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<tr>
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<td>Factors of 30 are <b>1, 2, 3, 5, 6, 10, 15 </b>and<b> 30</b></td>
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</tr>
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</tbody></table>
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</div>
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<p>Then the <span class="hilite"><b>common</b></span><b> factors</b> are those that are found in both
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lists:</p>
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<ul>
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<li>Notice that <b>1, 2, 3</b> and <b>6</b> appear in both lists?</li>
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<li>So, the <b>common factors</b> of 12 and 30 are: <b>1, 2, 3 </b>and<b> 6</b></li>
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</ul>
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<div class="def">
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<p>It is a <i>common</i> factor when it is a factor of two (or more) numbers.</p>
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</div>
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<p> </p>
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<p>Here is another example with three numbers:</p>
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<div class="example">
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<h3>Example: The common factors of 15, 30 and 105</h3>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td>Factors of 15 are <b>1, 3, 5, </b>and<b> 15</b></td>
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</tr>
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<tr>
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<td>Factors of 30 are <b>1, 2, 3, 5, 6, 10, 15 </b>and<b> 30</b></td>
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</tr>
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<tr>
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<td>Factors of 105 are <b>1, 3, 5, 7, 15, 21, 35 </b>and<b> 105</b></td>
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</tr>
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</tbody></table>
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<p>The factors that are common to all three numbers are <b>1, 3, 5</b> and <b>15</b></p>
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<p>In other words, the <b>common factors</b> of 15, 30 and 105 are <b>1, 3, 5</b> and <b>15</b></p>
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</div>
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<h2>What is the "Greatest Common Factor" ?</h2>
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<p>It is simply the <b>largest</b> of the common factors.</p>
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<p>In our previous example, the largest of the common factors is 15, so the <b>Greatest Common Factor</b> of 15, 30 and 105 is <b>15</b></p>
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<div class="def">
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<p>The "Greatest Common Factor" is the largest of the common factors (of two or more numbers)</p>
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</div>
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<h2>Why is this Useful?</h2>
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<p>One of the most useful things is when we want to simplify a fraction:</p>
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<div class="example">
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<h3>Example: How can we simplify <b> <span class="intbl"><em>12</em><strong>30</strong></span> </b>?</h3>
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</div>
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<p>Earlier we found that the Common Factors of 12 and 30 are 1, 2, 3 and 6, and so the <b>Greatest Common Factor is 6</b>.</p>
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<p>So the <b>largest</b> number we can divide both 12 and 30 exactly by is <b>6</b>, like this:</p>
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<table align="center" cellspacing="3" cellpadding="0" border="0">
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<tbody>
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<tr style="text-align:center;">
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<td> </td>
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<td style="width:60px;">÷ 6</td>
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<td> </td>
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</tr>
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<tr style="text-align:center;">
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<td colspan="3"><img src="images/style/right-over-arrow.gif" alt="right over arrow" height="25" width="75"> </td>
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</tr>
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<tr class="larger" align="center">
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<td><span class="intbl"><em>12</em><strong>30</strong></span></td>
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<td> = </td>
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<td><span class="intbl"><em>2</em><strong>5</strong></span></td>
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</tr>
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<tr style="text-align:center;">
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<td colspan="3"><img src="images/style/right-under-arrow.gif" alt="right under arrow" height="25" width="75"> </td>
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</tr>
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<tr style="text-align:center;">
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<td> </td>
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<td>÷ 6</td>
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<td> </td>
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</tr>
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</tbody></table>
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<div class="example">
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<p>The Greatest Common Factor of 12 and 30 is <b>6</b>.</p>
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<p>And so <b> <span class="intbl"><em>12</em><strong>30</strong></span> </b> can be simplified to <b> <span class="intbl"><em>2</em><strong>5</strong></span></b></p>
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</div>
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<h2>Finding the Greatest Common Factor</h2>
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<p>Here are three ways:</p>
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<div class="dotpoint">
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<p><b>1.</b> We can:</p>
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<ul>
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<li>find all <b>factors</b> of both numbers (use the <a href="numbers/factors-all-tool.html">All Factors Calculator</a>),</li>
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<li>then find the ones that are <b>common</b> to both, and</li>
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<li>then choose the <b>greatest</b>.</li>
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</ul>
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<p>Example:</p>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<th>Two Numbers</th>
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<th>Factors</th>
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<th>Common Factors</th>
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<th>Greatest<br>
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Common Factor</th>
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<th><i>Example Simplified<br>
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Fraction</i></th>
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</tr>
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<tr>
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<td style="text-align:center;"> <span class="intbl">9 and 12</span></td>
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<td> <b>9</b>: 1, 3, 9<br>
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<b>12</b>: 1, 2, 3, 4, 6, 12</td>
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<td style="text-align:center;">1, 3</td>
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<td style="text-align:center;"> <b>3</b></td>
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<td style="text-align:center;"> <span class="intbl"><em>9</em><strong>12</strong></span> = <span class="intbl"><em>3</em><strong>4</strong></span></td>
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</tr>
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</tbody></table>
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</div>
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<div class="simple">
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<p>And
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another example:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<th>Two Numbers</th>
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<th>Factors</th>
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<th>Common Factors</th>
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<th>Greatest<br>
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Common Factor</th>
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<th><i>Example Simplified<br>
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Fraction</i></th>
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</tr>
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<tr>
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<td style="text-align:center;">6 and 18</td>
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<td> <b>6</b>: 1, 2, 3, 6<br>
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<b>18</b>: 1, 2, 3, 6, 9, 18</td>
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<td style="text-align:center;">1, 2, 3, 6</td>
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<td style="text-align:center;"><b>6</b></td>
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<td style="text-align:center;"><span class="intbl"><em>6</em><strong>18</strong></span> = <span class="intbl"><em>1</em><strong>3</strong></span></td>
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</tr>
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</tbody></table>
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</div>
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</div>
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<p> </p>
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<div class="dotpoint">
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<p><b>2</b>. Or we can find the <a href="prime-factorization.html">prime factors</a> and combine the common ones together:</p>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<th>Two Numbers</th>
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<th>Thinking ...</th>
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<th>Greatest<br>
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Common Factor</th>
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<th><i>Example Simplified<br>
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Fraction</i></th>
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</tr>
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<tr style="text-align:center;">
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<td> 24 and 108</td>
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<td><b>2</b> × <b>2</b> × 2 × <b>3</b> = 24, and<br>
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<b>2</b> × <b>2</b> × <b>3</b> × 3 × 3 = 108</td>
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<td> 2 × 2 × 3 = <b>12</b></td>
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<td><span class="intbl"><em>24</em><strong>108</strong></span> = <span class="intbl"><em>2</em><strong>9</strong></span></td>
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</tr>
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</tbody></table>
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</div>
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</div><p> </p>
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<div class="dotpoint">
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<p><b>3.</b> Or sometimes we can just <b>play around</b> with the factors until we discover it:</p>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<th>Two Numbers</th>
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<th>Thinking ...</th>
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<th>Greatest<br>
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Common Factor</th>
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<th><i>Example Simplified<br>
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Fraction</i></th>
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</tr>
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<tr style="text-align:center;">
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<td><span class="intbl">9 and 12</span></td>
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<td><b>3</b> × 3 = 9 and <b>3</b> × 4 = 12</td>
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<td><b>3</b></td>
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<td><span class="intbl"><em>9</em><strong>12</strong></span> = <span class="intbl"><em>3</em><strong>4</strong></span></td>
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</tr>
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</tbody></table>
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</div>
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<p>But in that case we must check that we have found the <b>greatest</b> common factor.</p>
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</div>
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<h2>Greatest Common Factor Calculator</h2>
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<p>OK, there is also a really <i>easy</i> method: we can use the <a href="greatest-common-factor-tool.html">Greatest Common Factor Calculator</a> to find it automatically.</p>
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<h2>Other Names</h2>
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<p>The "Greatest Common Factor" is often abbreviated to "<b>GCF</b>", and is also known as:</p>
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<ul>
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<li>the "Greatest Common Divisor (GCD)", or</li>
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<li>the "Highest Common Factor (HCF)"</li>
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</ul>
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<p> </p>
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<div class="questions">
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<script>getQ(920,921,1382,1383,3548,3549,3550,3551,5010,5011);</script>
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</div>
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<div class="related">
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<a href="numbers/factors-all-tool.html">All Factors Calculator</a>
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<a href="fractions.html">Introduction to Fractions</a>
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<a href="greatest-common-factor-tool.html">Greatest Common Factor Calculator</a>
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<a href="simplifying-fractions.html">Simplifying Fractions</a>
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<a href="equivalent_fractions.html">Equivalent Fractions</a>
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<a href="fractions_addition.html">Adding Fractions</a>
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<a href="fractions_subtraction.html">Subtracting Fractions</a>
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<a href="fractions-menu.html">Fractions Index</a>
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