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417 lines
15 KiB
HTML
417 lines
15 KiB
HTML
<!DOCTYPE html>
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<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
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<!-- Mirrored from www.mathsisfun.com/prime-composite-number.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:35:36 GMT -->
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<title>Prime and Composite Numbers</title>
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<meta name="keywords" content="math, maths, mathematics, school, homework, education">
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</script>
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</head>
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<body id="bodybg">
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<div id="stt"></div>
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<div id="adTop"></div>
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<header>
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<div id="hdr"></div>
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<div id="tran"></div>
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<div id="adHide"></div>
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<div id="cookOK"></div>
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</header>
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<div class="mid">
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<nav>
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<div id="menuWide" class="menu"></div>
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<div id="logo"><a href="index.html"><img src="images/style/logo.svg" alt="Math is Fun"></a></div>
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<div id="search" role="search"></div>
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<div id="menuSlim" class="menu"></div>
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<div id="menuTiny" class="menu"></div>
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</nav>
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<div id="extra"></div>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Prime Numbers and Composite Numbers</h1>
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<div class="video">jpMYfW9XziU</div>
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<p class="center">A Prime Number is:</p>
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<div class="def">
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<p class="center">a whole number above 1 that <b>cannot</b> be made by multiplying other whole numbers</p>
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</div>
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<div class="example">
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<h3>Example: 5 is a <b>prime</b> number.</h3>
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<p>We cannot multiply other whole numbers like 2, 3 or 4 together to make 5</p>
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</div>
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<div class="example">
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<h3>Example: 6 is <b>not</b> a prime number</h3>
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<p>6 can be made by 2×3 so is NOT a prime number, it is a <b>composite number</b></p>
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</div>
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<h2>Not 1</h2>
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<p>Years ago 1 was included as a Prime, but now <b>it is not</b>:</p>
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<div class="def">
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<p class="large center">1 is <b>not Prime</b> and also <b>not Composite</b>.</p>
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</div>
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<h2 align="left">Dividing Into Equal Groups</h2>
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<p>It is all about trying to divide the number into equal groups</p>
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<p class="center large">Some <a href="whole-numbers.html">whole numbers</a> can be divided up exactly, and some can't!</p>
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<div class="example">
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<h3>Example: 6</h3>
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<p><span class="large">6</span> can be divided exactly by 2, or by 3:</p>
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<p class="center"><span class="large">6 = 2 × 3</span></p>
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<p>Like this:</p>
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<table align="center" width="80%" border="0">
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<tbody>
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<tr style="text-align:center;">
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<td><img src="numbers/images/composite-6-2.gif" alt="6 divided into 2" height="70" width="114"></td>
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<td>or</td>
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<td><img src="numbers/images/composite-6-3.gif" alt="6 divided into 3" height="70" width="114"></td>
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</tr>
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<tr style="text-align:center;">
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<td>
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<p>divided into 2 groups</p></td>
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<td> </td>
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<td>
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<p>divided into 3 groups</p></td>
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</tr>
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</tbody></table>
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<p> </p>
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<h3>Example: 7</h3>
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<p>But <span class="large">7</span> cannot be divided up exactly:</p>
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<p class="center"><img src="numbers/images/prime-7.gif" alt="7 is Prime" height="65" width="156"></p>
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</div>
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<p>And we give them names:</p>
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<ul>
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<div class="bigul">
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<li>When a number can be divided up exactly it is a <b>Composite Number</b></li>
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<li>When a number <b>cannot</b> be divided up exactly it is a <b>Prime Number</b></li>
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</div>
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</ul>
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<p class="center">So <b>6</b> is Composite, but <b>7</b> is Prime</p>
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<p>Like this:</p>
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<p class="center"><img itemprop="image" src="numbers/images/prime-composite.svg" alt="Prime vs Composite Number"></p>
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<p> </p>
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<p><b>And that explains it ... but there are some more details ...</b></p>
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<p> </p>
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<h2>Not Into Fractions</h2>
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<p>We are only dealing with whole numbers here! We are not going to cut things into halves or quarters.</p>
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<h2>Not Into Groups of 1</h2>
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<p>OK, we <b>could</b> have divided 7 into seven 1s (or one 7) like this:</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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<td><img src="numbers/images/prime-1x7.gif" alt="prime 1x7" height="24" width="180"></td>
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</tr>
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<tr>
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<td>
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<div class="center"><span class="large">7 = 1 x 7</span></div></td>
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</tr>
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</tbody></table>
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<p class="center larger">But we could do that for <b>any</b> whole number!</p>
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<p>So we are only interested in dividing by whole numbers <b>other than</b> the number itself.</p>
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<div class="example">
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<h3>Example: is<span class="large"> 7</span> a Prime Number or Composite Number?</h3>
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<p class="center"><img src="numbers/images/prime-7.gif" alt="7 is Prime" height="65" width="156"></p>
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<ul>
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<li>We <b>cannot</b> divide 7 exactly by 2 (we get 2 lots of 3, with one left over)</li>
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<li>We <b>cannot</b> divide 7 exactly by 3 (we get 3 lots of 2, with one left over)</li>
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<li>We <b>cannot</b> divide 7 exactly by 4, or 5, or 6.</li>
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</ul>
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<p>We can <b>only</b> divide 7 into one group of 7 (or seven groups of 1):</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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<td><img src="numbers/images/prime-1x7.gif" alt="prime 1x7" height="24" width="180"></td>
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</tr>
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<tr>
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<td>
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<div class="center"><span class="large">7 = 1 x 7</span></div></td>
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</tr>
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</tbody></table>
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<p>So 7 is a <b>Prime Number</b></p>
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</div>
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<p>And also:</p>
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<div class="def">
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<p class="center large">It is a <b>Composite Number</b> when it <b>can</b> be divided exactly
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by a whole number other than itself.</p>
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</div>
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<p>Like this:</p>
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<div class="example">
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<h3>Example: is<span class="large"> 6</span> a Prime Number or Composite Number?</h3>
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<p>6 can be divided exactly by 2, or by 3, as well as by 1 or 6:</p>
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<p class="center"><span class="large">6 = 1 × 6<br>
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6 = 2 × 3</span></p>
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<p>So 6 is a <b>Composite Number</b></p>
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</div>
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<p>Sometimes a number can be divided exactly in <b>many ways</b>:</p>
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<div class="example">
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<h3>Example: <b>12</b> can be divided exactly by 1, 2, 3, 4, 6 and 12:</h3>
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<p class="center large">1 × 12 = 12<br>
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2 × 6 = 12<br>
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3 × 4 = 12</p>
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<p>So 12 is a <b>Composite Number</b></p>
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</div>
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<p>And note this:</p>
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<div class="def">
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<p class="large">Any whole number greater than 1 is either <b>Prime</b> or <b>Composite</b></p>
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</div>
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<h2>Activity</h2>
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<div class="activity">
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You can try this <a href="numbers/prime-number-activity.html">Prime Numbers Activity</a>.
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</div>
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<h2>Factors</h2>
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<p>We can also define a Prime Number using factors.</p>
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<p class="center"><img src="numbers/images/factor-2x3.svg" alt="factor 2x3=6"><br>
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"Factors" are numbers we multiply<br>
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together
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to get
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another number.</p>
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<p>And we have:</p>
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<div class="def">
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<p class="center large">When <b>the only two factors</b> of a number are<b> 1 and the number</b>,<br>
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then it is a <b>Prime Number</b></p>
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</div>
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<p>It means the same as our previous definition, just stated using factors.</p>
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<p>And remember this is only about <a href="whole-numbers.html">Whole Numbers</a> (1, 2, 3, ... etc), not fractions or negative numbers. So don't say <i>"I could multiply ½ times 6 to get 3"</i>, OK?</p>
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<p>Examples:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;">
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<td style="width:250px;"><span class="larger">3 = 1 × 3 </span><br>
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(the only factors are 1 and 3)</td>
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<td class="larger">Prime</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 class="large"> </td>
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</tr>
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<tr style="text-align:center;">
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<td style="width:250px;"><span class="larger">6 = 1 × 6</span><br>
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<span class="larger">6 = 2 × 3<br>
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</span>(the factors are 1, 2, 3 and 6)</td>
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<td class="larger">Composite</td>
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</tr>
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</tbody></table>
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<h2>Examples From 1 to 14</h2>
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<p>Factors other than 1 or the number itself are <span class="hilite">highlighted</span>:</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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<td style="width:78px;">
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<div class="center"><b>Number</b></div></td>
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<td style="width:198px;">
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<div class="center"><b>Can be Exactly<br>
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Divided By</b></div></td>
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<td style="width:110px;">
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<div class="center"><b>Prime, or<br>
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Composite?</b> </div></td>
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</tr>
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<tr>
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<td style="width:78px;">
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<div class="center">1</div></td>
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<td colspan="2">
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<div class="center"><i>(1 is not prime or composite)</i></div></td>
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</tr>
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<tr>
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<td style="width:78px;">
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<div class="center">2</div></td>
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<td style="width:198px;">
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<div class="center">1, 2</div></td>
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<td style="width:110px;">
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<div class="center">Prime</div></td>
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</tr>
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<tr>
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<td style="width:78px;">
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<div class="center">3</div></td>
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<td style="width:198px;">
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<div class="center">1, 3</div></td>
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<td style="width:110px;">
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<div class="center">Prime</div></td>
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</tr>
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<tr>
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<td style="width:78px;">
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<div class="center">4</div></td>
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<td style="width:198px;">
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<div class="center">1, <span class="hilite">2</span>, 4</div></td>
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<td style="width:110px;">
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<div class="center">Composite</div></td>
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</tr>
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<tr>
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<td style="width:78px;">
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<div class="center">5</div></td>
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<td style="width:198px;">
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<div class="center">1, 5</div></td>
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<td style="width:110px;">
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<div class="center">Prime</div></td>
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</tr>
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<tr>
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<td style="width:78px;">
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<div class="center">6</div></td>
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<td style="width:198px;">
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<div class="center">1, <span class="hilite">2</span>, <span class="hilite">3</span>, 6</div></td>
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<td style="width:110px;">
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<div class="center">Composite</div></td>
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</tr>
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<tr>
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<td style="width:78px;">
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<div class="center">7</div></td>
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<td style="width:198px;">
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<div class="center">1, 7</div></td>
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<td style="width:110px;">
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<div class="center">Prime</div></td>
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</tr>
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<tr>
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<td style="width:78px;">
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<div class="center">8</div></td>
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<td style="width:198px;">
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<div class="center">1, <span class="hilite">2</span>, <span class="hilite">4</span>, 8</div></td>
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<td style="width:110px;">
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<div class="center">Composite</div></td>
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</tr>
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<tr>
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<td style="width:78px;">
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<div class="center">9</div></td>
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<td style="width:198px;">
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<div class="center">1, <span class="hilite">3</span>, 9</div></td>
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<td style="width:110px;">
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<div class="center">Composite</div></td>
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</tr>
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<tr>
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<td style="width:78px;">
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<div class="center">10</div></td>
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<td style="width:198px;">
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<div class="center">1, <span class="hilite">2</span>, <span class="hilite">5</span>, 10</div></td>
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<td style="width:110px;">
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<div class="center">Composite</div></td>
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</tr>
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<tr>
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<td>
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<div class="center">11</div></td>
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<td>
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<div class="center">1, 11</div></td>
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<td>
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<div class="center">Prime</div></td>
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</tr>
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<tr>
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<td>
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<div class="center">12</div></td>
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<td>
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<div class="center">1, <span class="hilite">2</span>, <span class="hilite">3</span>, <span class="hilite">4</span>, <span class="hilite">6</span>, 12</div></td>
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<td>
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<div class="center">Composite</div></td>
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</tr>
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<tr>
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<td>
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<div class="center">13</div></td>
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<td>
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<div class="center">1, 13</div></td>
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<td>
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<div class="center">Prime</div></td>
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</tr>
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<tr>
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<td>
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<div class="center">14</div></td>
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<td>
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<div class="center">1, <span class="hilite">2</span>, <span class="hilite">7</span>, 14</div></td>
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<td>
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<div class="center">Composite</div></td>
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</tr>
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<tr>
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||
<td>
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||
<div class="center">...</div></td>
|
||
<td>
|
||
<div class="center">...</div></td>
|
||
<td>
|
||
<div class="center">...</div></td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p>So when there are more factors than 1 or the number itself, the number is <b>Composite</b>.</p>
|
||
<p class="center"><i>A question for you: is 15 Prime or Composite?</i></p>
|
||
<h2>Why All the Fuss about Prime and Composite?</h2>
|
||
<p class="center">Because we can "break apart" Composite Numbers into Prime Number factors.</p>
|
||
<p style="float:left; margin: 0 10px 5px 0;"><img src="numbers/images/blocks223.svg" alt="stacked blocks labeled 2 2 and 3"></p>
|
||
<p>It is like the Prime Numbers are the <b>basic building blocks</b> of all numbers.</p>
|
||
<p>And the Composite Numbers are made up of Prime Numbers multiplied together.</p>
|
||
<p>Here we see it in action:</p>
|
||
<p class="center"><img src="numbers/images/prime-composite.svg" alt="prime composite"></p>
|
||
<div class="center larger">2 is Prime, 3 is Prime, 4 is Composite (=2×2), 5 is Prime, and so on...<br>
|
||
</div>
|
||
<div class="center larger"><br>
|
||
</div>
|
||
<div class="example">
|
||
<h3>Example: 12 is made by multiplying the prime numbers <b>2</b>, <b>2</b> and <b>3</b> together.</h3>
|
||
<p class="center large">12 = 2 × 2 × 3</p>
|
||
<p>The number <b>2</b> was repeated, which is OK.</p>
|
||
<p>In fact we can write it like this using the <a href="exponent.html">exponent</a> of 2:</p>
|
||
<p class="center"><span class="large">12 = 2<sup>2</sup> × 3</span></p>
|
||
</div>
|
||
<p> </p>
|
||
<div class="words">
|
||
<p>And that is why they are called "<b>Composite</b>" Numbers because composite means "something made by combining things"</p>
|
||
</div>
|
||
<p>This idea is so important it is called <a href="numbers/fundamental-theorem-arithmetic.html">The Fundamental Theorem of Arithmetic</a>.</p>
|
||
<p> </p>
|
||
<p>There are many puzzles in mathematics that can be solved more easily when we "break up" the Composite Numbers into their Prime Number factors.</p>
|
||
<p>And a lot of internet security is based on mathematics using prime numbers in a subject called <b>cryptography</b>.</p>
|
||
<p> </p>
|
||
<div class="questions">369, 1692, 1054, 1693, 2982, 2983, 2984, 3976, 2985, 3977</div>
|
||
|
||
<div class="related">
|
||
<a href="prime_numbers.html">Prime Numbers Chart and Calculator</a>
|
||
<a href="numbers/prime-properties.html">Prime Properties</a>
|
||
<a href="prime-factorization.html">Prime Factorization</a>
|
||
<a href="numbers/prime-number-activity.html">Prime Numbers Activity</a>
|
||
<a href="numbers/prime-factorization-tool.html">Prime Factorization Tool</a>
|
||
<a href="numbers/prime-numbers-advanced.html">Prime Numbers - Advanced</a>
|
||
</div>
|
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