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242 lines
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HTML
242 lines
9.1 KiB
HTML
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<title>Prime Factorization</title>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Prime Factorization</h1>
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<h2>Prime Numbers</h2>
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<p>A Prime Number is:</p>
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<div class="def">
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<p class="center">a whole number greater than 1 that can <b>not</b> be made by multiplying other whole numbers</p>
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</div>
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<p>The first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19 and 23,
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and we have a <a href="prime_numbers.html">prime number chart</a> if you need more.</p>
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<p>If we <b>can</b> make it by multiplying other whole numbers it is a <b>Composite Number</b>.</p>
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<p>Like this:</p>
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<p class="center"><img src="numbers/images/prime-composite.svg" alt="prime composite" height="155" width="532" ></p>
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<div class="center larger">2 is Prime, 3 is Prime, 4 is Composite (=2×2), 5 is Prime, and so on... </div>
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<h2>Factors</h2>
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<p>"Factors" are the numbers you multiply together to get
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another number:</p>
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<p class="center"><img src="numbers/images/factor-2x3.svg" alt="factors 2x3=6" height="70" width="181" ></p>
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<h2>Prime Factorization</h2>
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<div class="center80">
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<p class="larger">"Prime Factorization" is finding <b>which prime numbers</b> multiply together to make the original number.</p>
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</div>
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<p>Here are some examples:</p>
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<div class="example">
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<h3>Example: What are the prime factors of 12 ?</h3>
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<p>It is best to start working from the smallest prime number, which
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is 2, so let's check:</p>
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<p class="center larger">12 ÷ 2 = 6</p>
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<p>Yes, it divided exactly by 2. We have taken the first step!</p>
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<p>But 6 is not a prime number, so we need to go further. Let's try 2 again:</p>
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<p class="center larger">6 ÷ 2 = 3</p>
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<p>Yes, that worked also. And 3 <b>is</b> a prime number, so we have the answer:</p>
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<p class="center larger"><b>12 = 2 × 2 × 3</b></p>
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<p> </p>
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<p class="larger">As you can see, <b>every factor</b> is a <b>prime number</b>, so the answer must be right.</p>
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<p> </p>
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<p>Note: <b>12 = 2 × 2 × 3</b> can also be written using <a href="exponent.html">exponents</a> as <b>12 = 2<sup>2</sup> × 3</b></p>
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</div>
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<div class="example">
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<h3>Example: What is the prime factorization of 147 ?</h3>
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<p>Can we divide 147 exactly by 2?</p>
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<p class="center larger">147 ÷ 2 = 73½</p>
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<p>No it can't. The answer should be a whole number, and <span class="larger">73½</span> is not.</p>
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<p>Let's try the next prime
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number, 3:</p>
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<p class="center larger">147 ÷ 3 = 49</p>
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<p>That worked, now we try factoring 49.</p>
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<p>The next prime, 5, does not work. But 7 does, so we get:</p>
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<p class="center larger">49 ÷ 7 = 7</p>
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<p>And that is as far as we need to go, because all the factors are
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prime numbers.</p>
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<p class="center larger"><b>147 = 3 × 7 × 7</b></p>
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<p class="center">(or <b>147 =
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3 × 7<sup>2</sup></b> using exponents)</p>
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</div>
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<div class="example">
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<h3>Example: What is the prime factorization of 17 ?</h3>
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<p>Hang on ... <b>17 is a Prime Number</b>.</p>
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<p>So that is as far as we can go.</p>
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<p class="center"><span class="larger"><b>17 =
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17</b></span></p>
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</div>
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<h2>Another Method</h2>
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<p>We showed you how to do the factorization by starting at the smallest prime and working upwards.</p>
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<p>But sometimes it is easier to break a number down into <b>any factors</b> you can ... then work those factor down to primes.</p>
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<div class="example">
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<h3>Example: What are the prime factors of 90 ?</h3>
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<p>Break 90 into 9 × 10</p>
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<ul>
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<li>The prime factors of 9 are <b>3 and 3</b></li>
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<li>The prime factors of 10 are <b>2 and 5</b></li>
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</ul>
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<p>So the prime factors of 90 are <b>3, 3, 2 and 5</b></p>
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</div>
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<h2>Factor Tree</h2>
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<p>And a "Factor Tree" can help: find <b>any factors</b> of the number, then the factors of those numbers, etc, until we can't factor any more.</p>
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<div class="example">
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<h3>Example: 48</h3>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="numbers/images/factor-tree-48.svg" alt="factor tree 48 = 2 x 2 x 2 x 2 x 3" height="253" width="229" ></p>
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<p><b>48 = 8 × 6</b>, so we write down "8" and "6" below 48</p>
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<p>Now we continue and factor 8 into <b>4 × 2</b></p>
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<p>Then 4 into <b>2 × 2</b></p>
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<p>And lastly 6 into <b>3 × 2</b></p>
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<p> </p>
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<p>We can't factor any more, so we have found the prime factors.</p>
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<p>Which reveals that <b>48 = 2 × 2 × 2 × 2 × 3</b></p>
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<p>(or <b>48 =
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2<sup>4</sup> × 3</b> using exponents)</p>
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</div>
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<h2>Why find Prime Factors?</h2>
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<p>A prime number can only be divided by 1 or itself, so it cannot
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be factored any further!</p>
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<p>Every other whole number can be broken down into prime number factors.</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><span class="large"><img src="numbers/images/blocks223.svg" alt="blocks 2 2 3" height="125" width="149" ></span></td>
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<td> </td>
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<td>
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<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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</tbody></table>
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<p>This idea can be very useful when working with big numbers, such as in Cryptography.</p>
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<h2>Cryptography</h2>
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<p>Cryptography is the study of secret codes. Prime Factorization is very important to people who try
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to make (or break) secret codes based on numbers.</p>
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<p>That is because factoring very large numbers is very hard, and can take computers a long time to do.</p>
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<p>If you want to
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know more, the subject is "encryption" or "cryptography".</p>
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<h2>Unique</h2>
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<p>And here is another thing:</p>
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<p class="center"><b>There is only one (unique!) set of prime factors for any number.</b></p>
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<div class="example">
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<h3>Example: the prime factors of 330 are 2, 3, 5 and 11</h3>
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<p class="center"><span class="large">330 = 2 × 3 × 5</span><span class="large"> × 11</span></p>
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<p>There is no other possible set of prime numbers that can be multiplied to make 330.</p>
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</div>
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<p>In fact this idea is so important it is called the <b><a href="numbers/fundamental-theorem-arithmetic.html">Fundamental Theorem of Arithmetic</a></b>.</p>
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<h2>Prime Factorization Tool</h2>
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<p>OK, we have one more method ... use our <a href="numbers/prime-factorization-tool.html">Prime Factorization Tool</a> that can work out the prime factors for numbers up to
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4,294,967,296.</p>
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<p> </p>
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<div class="questions">370, 1055, 1694, 1695, 1696, 1697</div>
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<div class="related">
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<a href="prime-composite-number.html">Prime and Composite Numbers</a>
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<a href="prime_numbers.html">Prime Numbers Chart</a>
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<a href="numbers/prime-factorization-tool.html">Prime Factorization Tool</a>
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<a href="divisibility-rules.html">Divisibility Rules</a>
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</div>
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<!-- #EndEditable -->
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</article>
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