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<title>Fundamental Theorem of Arithmetic</title>
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<h1 align="center">Fundamental Theorem of Arithmetic</h1>
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<h2>The Basic Idea</h2>
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<p>The <b>Basic Idea</b> is that any <a href="../whole-numbers.html">integer</a> above 1 is either a <a href="../prime-composite-number.html">Prime Number</a>, or can be made by <b>multiplying Prime Numbers</b> together. Like this:</p>
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<p class="center"><img src="images/prime-composite.svg" alt="prime (2,3,5,7,...) vs composite (4=2x2, 6=2x3, 8=2x2x2, ...)" /></p>
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<p>This continues on:</p>
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<ul>
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<li>10 is 2×5</li>
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<li>11 is Prime,</li>
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<li>12 is 2×2×3</li>
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<li>13 is Prime</li>
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<li>14 is 2×7</li>
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<li>15 is 3×5</li>
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<li>16 is 2×2×2×2</li>
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<li>17 is Prime</li>
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<li>etc...</li>
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</ul>
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<p>So they are either <b>prime</b>, or <b>primes multiplied together</b></p>
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<p>Read on for an explanation ...</p>
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<h2>The Fundamental Theorem of Arithmetic</h2>
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<p>Let us start with the definition:</p>
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<div class="def">
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<p>Any integer greater than 1 is either a <b>prime number</b>, or can be written as a <b>unique product of prime numbers</b> (ignoring the order).</p>
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</div>
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<h2>What does it mean?</h2>
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<p>Let's build up the ideas piece by piece:</p>
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<div class="dotpoint">
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<p>"Any <a href="../whole-numbers.html">integer</a> greater than 1" means the numbers <b>2, 3, 4, 5, 6, ...</b> etc.</p>
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</div>
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<div class="dotpoint">
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<p>A <a href="../prime-composite-number.html">Prime Number</a> is a number that cannot be exactly divided by any other number (except 1 or itself). </p>
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</div>
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<div class="def">
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<p>The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, ... (<a href="../prime_numbers.html">and more</a>)</p>
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</div>
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<div class="dotpoint">
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<p>"...product of prime numbers" means that we <b>multiply prime numbers together</b>.</p>
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<p>So, by multiplying prime numbers we can create any other whole number.</p>
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</div>
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<div class="example">
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<h3>Example: 42 </h3>
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<p>Can we make 42 by multiplying <b>only prime numbers?</b> Let's see:</p>
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<p align="center" class="large">2 × 3 × 7 = 42</p>
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<p>Yes, <b>2</b>, <b>3</b> and <b>7</b> are prime numbers, and when multiplied together they make <b>42</b>.</p>
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<p> Try some other examples for yourself. How about 30? Or 33?</p>
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</div>
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<table border="0" align="center">
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<tr>
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<td><span class="large"><img src="images/blocks223.svg" alt="2 and 2 and 3 " /></span></td>
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<td><p>It is like the Prime Numbers are the <b>basic building blocks</b> of all numbers.</p></td>
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</tr>
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</table>
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<div class="dotpoint">
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<p>"... <b>unique</b> product of prime numbers" means there is only one (unique!) set of prime numbers that will work</p>
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</div>
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<div class="example">
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<h3>Example: we just showed that 42 is made by the prime numbers <b>2</b>, <b>3</b> and <b>7</b>:</h3>
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<p align="center" class="large">2 × 3 × 7 = 42</p>
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<p><b>No other prime numbers will work!</b></p>
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<p>We could try <span class="large">2 × 3 × 5, </span>or<span class="large"> 5 × 11, </span>but none of them will work:</p>
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<p align="center"><span class="large">Only 2, 3 and 7 make 42</span></p>
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</div>
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<p> </p>
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<h2>So there you have it! </h2>
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<div class="def">
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<p>Any of the numbers <b>2, 3, 4, 5, 6, ...</b> etc are either prime numbers, or can be made by multiplying prime numbers together. </p>
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<p>And there is only one (unique) set of prime numbers that works in each case.</p>
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</div>
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<p>More examples:</p>
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<div class="example">
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<h3>Example: 7 </h3>
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<p>7 is already a prime number</p>
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</div>
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<div class="example">
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<h3>Example: 22 </h3>
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<p>22 can be made by multiplying the prime numbers <b>2</b><b> </b>and <b>11</b> together.</p>
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<p align="center" class="large"> 2 × 11 = 22</p>
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<p>No other combination of prime numbers will work.</p>
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</div>
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<h2>Ignore the Order</h2>
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<p>Also, at the top I said "ignoring the order". By that I mean:</p>
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<ul>
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<li><b>2 × 11 = 22</b> is the same as </li>
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<li><b>11 × 2 = 22</b></li>
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</ul>
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<p>So don't just rearrange the numbers and say "it isn't unique", OK?</p>
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<h2>Repeated Numbers</h2>
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<p>We may have to repeat a prime number!</p>
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<div class="example">
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<h3>Example: 12 is made by multiplying the prime numbers <b>2</b>, <b>2</b> and <b>3</b> together.</h3>
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<p align="center" class="large">12 = 2 × 2 × 3</p>
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<p>That is OK. In fact we can write it like this:</p>
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<p align="center"><span class="large">12 = 2<sup>2</sup> × 3</span></p>
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<p>It is still a <b>unique combination</b> (2, 2 and 3)</p>
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<p>(Note: <b>4 × 3</b> does not work, as 4 is not a prime number)</p>
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</div>
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<h2>The First Few</h2>
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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 width="78"><div align="center">2</div></td>
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<td width="198"><div align="center">Is a Prime</div></td>
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</tr>
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<tr>
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<td width="78"><div align="center">3</div></td>
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<td width="198"><div align="center">Is a Prime</div></td>
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</tr>
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<tr>
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<td width="78"><div align="center">4</div></td>
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<td width="198"><div align="center" class="large">= 2×2 = 2<sup>2</sup></div></td>
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</tr>
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<tr>
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<td width="78"><div align="center">5</div></td>
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<td width="198"><div align="center">Is a Prime</div></td>
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</tr>
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<tr>
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<td width="78"><div align="center">6</div></td>
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<td width="198"><div align="center"><span class="large">= 2</span><span class="large">×</span><span class="large">3</span></div></td>
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</tr>
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<tr>
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<td width="78"><div align="center">7</div></td>
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<td width="198"><div align="center">Is a Prime</div></td>
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</tr>
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<tr>
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<td width="78"><div align="center">8</div></td>
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<td width="198"><div align="center" class="large">= 2×2×2 = 2<sup>3</sup></div></td>
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</tr>
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<tr>
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<td width="78"><div align="center">9</div></td>
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<td width="198"><div align="center" class="large">= 3×3 = 3<sup>2</sup></div></td>
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</tr>
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<tr>
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<td width="78"><div align="center">10</div></td>
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<td width="198"><div align="center"><span class="large">= 2</span><span class="large">×</span><span class="large">5</span></div></td>
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</tr>
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<tr>
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<td><div align="center">11</div></td>
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<td><div align="center">Is a Prime</div></td>
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</tr>
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<tr>
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<td><div align="center">12</div></td>
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<td><div align="center" class="large">= 2×2×3 = 2<sup>2</sup>×3</div></td>
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</tr>
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<tr>
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<td><div align="center">13</div></td>
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<td><div align="center">Is a Prime</div></td>
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</tr>
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<tr>
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<td><div align="center">14</div></td>
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<td><div align="center"><span class="large">= 2</span><span class="large">×</span><span class="large">7</span></div></td>
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</tr>
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<tr>
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<td><div align="center">...</div></td>
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<td><div align="center">...</div></td>
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</tr>
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</table></div>
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<p class="center"><i>Why not continue this list to 100 yourself?</i></p>
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<h2>Summary</h2>
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<p align="center"><span class="larger">The Fundamental Theorem of Arithmetic is like a "guarantee"<br>
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that any integer greater than 1 <br>
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is either prime<br>
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or
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can be made
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by multiplying prime numbers</span></p>
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<p align="center"><i>and</i></p>
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<p align="center" class="larger">There is only one way to do that in each case</p>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(424, 425, 1682, 1056, 1057, 1683, 2986, 2987, 3978, 3979);</script>
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</div>
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<div class="related">
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<a href="../prime-composite-number.html">Prime and
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Composite Numbers</a> <a href="../prime_numbers.html">Prime Numbers
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Chart</a> <a href="../prime-factorization.html">Prime Factorization</a> </div>
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Copyright © 2017 MathsIsFun.com
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