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155 lines
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HTML
155 lines
7.8 KiB
HTML
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<title>Least Common Multiple</title>
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<meta name="description" content="The smallest positive number that is a multiple of two or more numbers" />
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<h1 align="center">Least Common Multiple</h1>
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<p align="center"><i>The smallest positive number that is a multiple of two or more numbers.</i></p>
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<h2>Let's start with an Example ... </h2>
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<div class="example">
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<h3><b>Least Common Multiple of 3 and 5:</b></h3>
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<div class="dotpoint">
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<p>List the<b> Multiples </b>of each number,</p>
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<p>The multiples of <b>3</b> are 3, 6, 9, 12, 15, 18, ... etc<br>
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The multiples of <b>5</b> are 5, 10, 15, 20, 25, ... etc</p>
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</div>
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<div class="dotpoint">
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<p>Find the first <b>Common</b> (same) value:</p>
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<p class="center"><img src="numbers/images/multiples-3-5.svg" alt="LCM of 3 and 5 is 15" /></p>
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</div>
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<div style="clear:both"></div>
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<p class="larger">The <b>Least Common Multiple</b> of 3 and 5 is <b>15</b></p>
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<p>(15 is a multiple of both 3 and 5, and is the smallest number like that.)</p>
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</div>
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<h2>So ... what is a "Multiple" ?</h2>
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<p> We get a <b>multiple</b> of a number when we <b>multiply it by another number</b>. Such as multiplying by 1, 2, 3, 4, 5, etc, <i>but not zero</i>. Just like the multiplication table. <br />
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<br />
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Here are some examples: </p>
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<div class="simple">
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<table border="0" align="center">
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<tr>
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<td>The multiples of <b>4</b> are: 4,8,12,16,20,24,28,32,36,40,44,...</td>
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</tr>
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<tr>
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<td>The multiples of <b>5</b> are: 5,10,15,20,25,30,35,40,45,50,...</td>
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</tr>
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</table>
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</div>
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<h2>What is a "Common Multiple" ?</h2>
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<p> Say we have listed the first few multiples of <b>4</b> and <b>5</b>: the <b>common multiples</b> are those that are found in both
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lists:</p>
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<div class="simple">
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<table border="0" align="center">
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<tr>
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<td>The multiples of 4 are: 4,8,12,16,<b>20</b>,24,28,32,36,<b>40</b>,44,...</td>
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</tr>
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<tr>
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<td>The multiples of 5 are: 5,10,15,<b>20</b>,25,30,35,<b>40</b>,45,50,...</td>
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</tr>
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</table>
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</div>
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<p align="center"><span class="larger">Notice that <b>20</b> and <b>40</b> appear in both lists? </span><br />
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So, the common multiples of 4 and 5 are: <b>20, 40,</b> (and <b>60</b>, <b>80</b>, etc ..., too) </p>
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<h2>What is the "Least Common Multiple" ?</h2>
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<p align="center" class="larger">It is simply the <b>smallest</b> of the common multiples. </p>
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<p>In our previous example, the smallest of the common multiples is <b>20</b> ...</p>
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<p align="center">... so the <b>Least</b> Common Multiple of 4 and 5 is <b>20</b>. </p>
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<h2> Finding the Least Common Multiple</h2>
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<p>List the multiples of the numbers until we get our first match.</p>
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<div class="simple">
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<div class="example">
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<h3>Example: Find the least common multiple of 4 and 10:</h3>
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<p>The multiples of 4 are: <b><i>4, 8, 12, 16, 20, ...</i></b><br />
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and the multiples of 10 are: <b><i> 10, 20, ...</i></b></p>
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<p>Aha! there is a match at 20. It looks like this:</p>
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<p class="center"><img src="numbers/images/multiples-4-10.svg" alt="LCM of 4 and 10 is 20" /> </p>
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<p>So the least common multiple of 4 and 10 is <b>20</b></p>
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</div>
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<div class="example">
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<h3>Example: Find the least common multiple of 6 and 15:</h3>
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<p>The multiples of 6 are: <b><i>6, 12, 18, 24, 30, ...</i></b><br />
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and the multiples of 15 are: <b><i> 15, 30, ...</i></b></p>
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<p>There is a match at 30</p>
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<p>So the least common multiple of 6 and 15 is <b>30</b> </p>
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</div>
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</div>
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<h2>More than 2 Numbers</h2>
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<p>We can also find the least common multiple of three (or more) numbers.</p>
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<div class="example">
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<h3>Example: Find the least common multiple of 4, 6, and 8</h3>
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<p>Multiples of 4 are: 4, 8, 12, 16, 20, <b>24</b>, 28, 32, 36, ...<br />
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Multiples of 6 are: 6, 12, 18, <b>24</b>, 30, 36, ...<br />
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Multiples of 8 are: 8, 16, <b>24</b>, 32, 40, ....</p>
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<p>So <b>24</b> is the least common multiple (I can't find a smaller one!) </p>
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</div>
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<p><i>Hint: We can have smaller lists for the bigger numbers.</i></p>
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<h2>Least Common Multiple Tool</h2>
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<p>There is <i>another</i> method: the <a href="least-common-multiple-tool.html">Least Common Multiple Tool</a> does it automatically. (Yes, we waited until the end to tell you!)</p>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(924, 925, 926, 927, 1393, 1394, 3556, 3557, 3558, 3559);</script> </div>
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<div class="related"> <a href="fractions.html">Introduction to Fractions</a> <a href="simplifying-fractions.html">Simplifying Fractions</a> <a href="comparing-fractions.html">Comparing Fractions</a> <a href="least-common-multiple-tool.html">Least Common Multiple Tool</a> <a href="least-common-denominator.html">Least Common Denominator</a> <a href="greatest-common-factor.html">Greatest Common Factor</a> <a href="fractions_addition.html">Adding Fractions</a> <a href="fractions_subtraction.html">Subtracting Fractions</a> <a href="fractions-menu.html">Fractions Index</a></div>
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