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165 lines
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165 lines
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<h1 class="center">Factors and Multiples</h1>
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<p>Factors and multiples are <b>different</b> things.</p>
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<p>But they both involve <b>multiplication</b>:</p>
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<ul>
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<li>Factors are what we can multiply to get the number</li>
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<li>Multiples are what we get <b>after</b> multiplying the number by an <a href="../whole-numbers.html">integer</a> (not a fraction).</li>
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</ul>
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<div class="example">
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<h3><span class="center">Example: the positive factors, and some multiples, of 6</span>:</h3>
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<p class="center"><img src="images/factors-multiples.svg" alt="factors of 6: ..., 1, 2, 3, 6; multiples of 6: ... 0, 6, 12, 18, ..."></p>
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<p>Factors:</p>
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<ul>
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<li>1 × 6 = 6, so <b>1</b> and <b>6</b> are factors of 6</li>
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<li>2 × 3 = 6, so <b>2</b> and <b>3</b> are factors of 6</li>
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</ul>
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<p>Multiples:</p>
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<ul>
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<li>0 × 6 = 0, so <b>0</b> is a multiple of 6</li>
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<li>1 × 6 = 6, so <b>6</b> is a multiple of 6</li>
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<li>2 × 6 = 12, so <b>12</b> is a multiple of 6</li>
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<li>and so on</li>
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</ul>
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<p>(Note: there are negative factors and multiples as well)</p>
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</div>
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<p>Here are the details:</p>
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<h2>Factors</h2>
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<p>"Factors" are the numbers we can <b>multiply together</b> to get
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another number:</p>
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<p class="center"><img src="images/factor-2x3.svg" alt="in 2x3=6, 2 and 3 are factors"><br>
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<br>
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<span class="larger">2 and 3 are factors of 6</span></p>
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<p>A number can have <b>many</b> factors.</p>
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<div class="example">
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<h3>Example: 12</h3>
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<ul>
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<li>3 × 4 = 12, so <b>3</b> and <b>4</b> are factors of 12</li>
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<li>Also 2 × 6 = 12, so <b>2</b> and <b>6</b> are also factors of 12,</li>
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<li>And 1 × 12 = 12, so <b>1</b> and <b>12</b> are factors of 12 as well.</li>
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</ul>
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<p>AND because <a href="../multiplying-negatives.html">multiplying negatives makes a positive</a>, −1, −2, −3, −4, −6 and −12 are also factors of 12:</p>
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<ul>
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<li>(−1) × (−12) = 12</li>
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<li>(−2) × (−6) = 12</li>
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<li>(−3) × (−4) = 12</li>
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</ul>
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<p>So ALL the factors of 12 are:</p>
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<p class="center"><b>1, 2, 3, 4, 6 and 12</b><br>
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AND <b>−1, −2, −3, −4, −6 and −12</b></p>
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</div>
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<p>Learn about <a href="../greatest-common-factor.html">Greatest Common Factor</a> and how to find <a href="factors-all-tool.html">All Factors of a Number</a>.</p>
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<h2>Multiples</h2>
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<p>A multiple is the result of <b>multiplying</b> a number <b>by an <a href="../whole-numbers.html">integer</a></b> (not a fraction).</p>
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<div class="example">
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<h3>Example: Multiples of 3:</h3>
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<p class="center"><img itemprop="image" src="images/multiples-3.svg" alt="Multiples of 3: ..., −9, −6, −3, 0, 3, 6, 9, ..."><br>
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..., −9, −6, −3, 0, 3, 6, 9, ...</p>
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<p>Example: 15 <b>is</b> a multiple of 3, as 3 × 5 = 15<br>
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Example: 16 is <b>not</b> a multiple of 3</p>
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</div>
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<div class="example">
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<h3>Example: Multiples of 5:</h3>
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<p class="center"><img itemprop="image" src="images/multiples-5.svg" alt="Multiples of 5: ..., −15, −10, −5, 0, 5, 10, 15, ..."><br>
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..., −15, −10, −5, 0, 5, 10, 15, ...</p>
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<p>Example: 10 <b>is</b> a multiple of 5, as 5 × 2 = 10<br>
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Example: 11 is <b>not</b> a multiple of 5</p>
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</div>
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Learn more about <a href="../least-common-multiple.html">Least Common Multiple</a>.
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<h3>Multiples of Anything</h3>
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<p>We must <b>multiply by an integer</b>, but the number that is being multiplied can be anything.</p>
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<div class="example">
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<h3>Example: Multiples of <a href="pi.html" title="Pi"><span class="times">π</span></a></h3>
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<p class="center">..., −2<span class="times">π</span>, −<span class="times">π</span>, 0, <span class="times">π</span>, 2<span class="times">π</span>, 3<span class="times">π</span>, 4<span class="times">π</span>, ...</p>
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</div>
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<p> </p>
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<div class="related">
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<a href="factors-all-tool.html">All Factors of a Number </a>
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<a href="factors-multiples-table.html">Table of Factors and Multiples</a>
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<a href="index.html">Numbers Index</a>
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