I never known maths can be this simple.
I love the explanation 

<h2>Finding the Prime Factorization of a Composite Number</h2>
<p>In the previous section, we found the factors of a number. Prime numbers have only two factors, the number \(1\) and the prime number itself. Composite numbers have more than two factors, and every composite number can be written as a unique product of primes. This is called the <strong>prime factorization</strong> of a number. When we write the prime factorization of a number, we are rewriting the number as a product of primes. Finding the prime factorization of a composite number will help you later on in your study of pre-algebra topics.</p>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Prime Factorization</h3>
<p>The prime factorization of a number is the product of prime numbers that equals the number.</p>
</div>
</div>
<div class="ns-tutorial">
<p>You may want to refer to the following list of prime numbers less than \(50\) as you work through this section.</p>
<h4>\(2,3,5,7,11,13,17,19,23,29,31,37,41,43,47\)</h4>
<h3 data-type="title">Prime Factorization Using the Factor Tree Method</h3>
<p>One way to find the prime factorization of a number is to make a <strong>factor tree</strong>. We start by writing the number, and then writing it as the product of two factors. We write the factors below the number and connect them to the number with a small line segment—a “branch” of the factor tree.</p>
<p>If a factor is prime, we circle it (like a bud on a tree), and do not factor that “branch” any further. If a factor is not prime, we repeat this process, writing it as the product of two factors and adding new branches to the tree.</p>
<p>We continue until all the branches end with a prime. When the factor tree is complete, the circled primes give us the prime factorization.</p>
<p>For example, let’s find the prime factorization of \(36.\) We can start with any factor pair such as \(3\) and \(12.\) We write \(3\) and \(12\) below \(36\) with branches connecting them.</p>
<img alt="The figure shows a factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end." src="https://data.ulearngo.com/assets/3ad17809-8867-4def-ae72-7109aaaa4c53"/>
<p>The factor \(3\) is prime, so we circle it. The factor \(12\) is composite, so we need to find its factors. Let’s use \(3\) and \(4.\) We write these factors on the tree under the \(12.\)</p>
<img alt="The figure shows a factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end. Two more branches are splitting out from under 12. The right branch has the number 4 at the end and the left branch has the number 3 at the end." src="https://data.ulearngo.com/assets/96076100-886d-4a7e-b3bc-b51055fe3817"/>
<p>The factor \(3\) is prime, so we circle it. The factor \(4\) is composite, and it factors into \(2·2.\) We write these factors under the \(4.\) Since \(2\) is prime, we circle both \(2\text{s}.\)</p>
<img alt="The figure shows a factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end. Two more branches are splitting out from under 12. The right branch has the number 4 at the end and the left branch has the number 3 at the end with a circle around it. Two more branches are splitting out from under 4. Both the left and right branch have the number 2 at the end with a circle around it." src="https://data.ulearngo.com/assets/b235944e-4c36-4080-8690-5fb4624f28fb"/>
<p>The prime factorization is the product of the circled primes. We generally write the prime factorization in order from least to greatest.</p>
<div data-label="">\(2\cdot 2\cdot 3\cdot 3\)</div>
<p>In cases like this, where some of the prime factors are repeated, we can write prime factorization in exponential form.</p>
<div data-label="">\(\begin{array}{c}2\cdot 2\cdot 3\cdot 3\\ \\ {2}^{2}\cdot {3}^{2}\end{array}\)</div>
<p>Note that we could have started our factor tree with any factor pair of \(36.\) We chose \(12\) and \(3,\) but the same result would have been the same if we had started with \(2\) and \(18,4\) and \(9,\text{or}\phantom{\rule{0.2em}{0ex}}6\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}6.\)</p>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Finding the prime factorization of a composite number using the tree method.</h3>
<ol type="1">
<li>Find any factor pair of the given number, and use these numbers to create two branches.</li>
<li>If a factor is prime, that branch is complete. Circle the prime.</li>
<li>If a factor is not prime, write it as the product of a factor pair and continue the process.</li>
<li>Write the composite number as the product of all the circled primes.</li>
</ol>
</div>
</div>
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<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Find the prime factorization of \(48\) using the factor tree method.</p>
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<div>
<h3 data-type="title">Solution</h3>
<table data-label="" summary="The figure shows multiple factor trees with the number 48 at the top. In the first tree two branches are splitting out from under 48. The branches use the factor pair 2 and 24 with 24 at the end of the right branch and 2 at the end of the left branch. Two has a circle around it to show that it is prime and that branch is complete. In the next tree the previous tree is repeated, but now with two branches splitting out from under 24. The branches use the factor pair 4 and 6 with 6 at the end of the right branch and 4 at the end of the left branch. Neither of these factors is circled because they are not prime. In the last tree the previous tree is repeated, but now with two branches splitting out from under 4 and two branches splitting out from under 6. The branches under 4 use the factor pair 2 and 2. Both of these two's are circled to show that they are prime and that branch is complete. The branches under 6 use the factor pair 2 and 3. Both of these numbers are circled to show that they are prime and that branch is complete. The prime factorization of the number 48 is made up of all of the circled numbers from the factor tree which is 2, 2, 2, 2, and 3. The prime factorization can be written as 2 times 2 times 2 times 2 times 3 or using exponents for repeated multiplication of 2 it can be written as 2 to the fourth power times 3.">
<tbody>
<tr>
<td>
<p>We can start our tree using any factor pair of 48. Let's use 2 and 24.</p>
<p>We circle the 2 because it is prime and so that branch is complete.</p>
</td>
<td><img alt="." src="https://data.ulearngo.com/assets/7fc12959-6a25-4f6c-b616-603acbf465c6"/></td>
</tr>
<tr>
<td>Now we will factor 24. Let's use 4 and 6.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/769026a4-6fd3-40a6-aa47-c8369f942286"/></td>
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<tr>
<td>
<p>Neither factor is prime, so we do not circle either.</p>
<div> </div>

We factor the 4, using 2 and 2.

<div> </div>

We factor 6, using 2 and 3.

<p> </p>
<p>We circle the 2s and the 3 since they are prime. Now all of the branches end in a prime.</p>
</td>
<td><img alt="." src="https://data.ulearngo.com/assets/d5bba3f3-6d67-4fd6-8497-a95f21c68366"/></td>
</tr>
<tr>
<td>Write the product of the circled numbers.</td>
<td>\(2\cdot 2\cdot 2\cdot 2\cdot 3\)</td>
</tr>
<tr>
<td>Write in exponential form.</td>
<td>\({2}^{4}\cdot 3\)</td>
</tr>
</tbody>
</table>
<p>Check this on your own by multiplying all the factors together. The result should be \(48.\)</p>
</div>
</div>
</div>
</div>
</div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Find the prime factorization of 84 using the factor tree method.</p>
</div>
<div>
<h3 data-type="title">Solution</h3>
<table data-label="" summary="The figure shows two factor trees with the number 84 at the top. In the first tree two branches are splitting out from under 84. The branches use the factor pair 4 and 21 with 21 at the end of the right branch and 4 at the end of the left branch. In the last tree the previous tree is repeated, but now with two branches splitting out from under 4 and two branches splitting out from under 21. The branches under 4 use the factor pair 2 and 2. Both of these two's are circled to show that they are prime and that branch is complete. The branches under 21 use the factor pair 3 and 7. Both of these numbers are circled to show that they are prime and that branch is complete. The prime factorization of the number 84 is made up of all of the circled numbers from the factor tree which is 2, 2, 3, and 7. The prime factorization can be written as 2 times 2 times 3 times 7 or using exponents for repeated multiplication of 2 it can be written as 2 squared times 3 times 7.">
<tbody>
<tr>
<td>
<p>We start with the factor pair 4 and 21.</p>
<p>Neither factor is prime so we factor them further.</p>
</td>
<td><img alt="." src="https://data.ulearngo.com/assets/2cd82762-e11d-425f-8c82-287b437f1e48"/></td>
</tr>
<tr>
<td>Now the factors are all prime, so we circle them.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/0e8d7b9e-4149-4abf-b9b6-8ec0398283c7"/></td>
</tr>
<tr>
<td>Then we write 84 as the product of all circled primes.</td>
<td>\(2\cdot 2\cdot 3\cdot 7\)

<div> </div>

\({2}^{2}\cdot 3\cdot 7\)</td>
</tr>
</tbody>
</table>
<p>Draw a factor tree of \(84.\)</p>
</div>
</div>
</div>
</div>
</div>
<h3 data-type="title">Prime Factorization Using the Ladder Method</h3>
<p>The <strong>ladder method</strong> is another way to find the prime factors of a composite number. It leads to the same result as the factor tree method. Some people prefer the ladder method to the factor tree method, and vice versa.</p>
<p>To begin building the “ladder,” divide the given number by its smallest prime factor. For example, to start the ladder for \(36,\) we divide \(36\) by \(2,\) the smallest prime factor of \(36.\)</p>
<p><img alt="The image shows the division of 2 into 36 to get the quotient 18. This division is represented using a division bracket with 2 on the outside left of the bracket, 36 inside the bracket and 18 above the 36, outside the bracket." src="https://data.ulearngo.com/assets/dc74d0ed-5abb-46b2-b367-b1ca5e0198a9"/></p>
<p>To add a “step” to the ladder, we continue dividing by the same prime until it no longer divides evenly.</p>
<p><img alt="The image shows the division of 2 into 36 to get the quotient 18. This division is represented using a division bracket with 2 on the outside left of the bracket, 36 inside the bracket and 18 above the 36, outside the bracket. Another division bracket is written around the 18 with a 2 on the outside left of the bracket and a 9 above the 18, outside of the bracket." src="https://data.ulearngo.com/assets/ae73c7b9-8394-4c53-b978-a7cb8f0135bc"/></p>
<p>Then we divide by the next prime; so we divide \(9\) by \(3.\)</p>
<p><img alt="The image shows the division of 2 into 36 to get the quotient 18. This division is represented using a division bracket with 2 on the outside left of the bracket, 36 inside the bracket and 18 above the 36, outside the bracket. Another division bracket is written around the 18 with a 2 on the outside left of the bracket and a 9 above the 18, outside of the bracket. Another division bracket is written around the 9 with a 3 on the outside left of the bracket and a 3 above the 9, outside of the bracket." src="https://data.ulearngo.com/assets/f2297a59-a208-4215-8add-1b2448523337"/></p>
<p>We continue dividing up the ladder in this way until the quotient is prime. Since the quotient, \(3,\) is prime, we stop here.</p>
<p>Do you see why the ladder method is sometimes called stacked division?</p>
<p>The prime factorization is the product of all the primes on the sides and top of the ladder.</p>
<div data-label="">\(\begin{array}{c}2\cdot 2\cdot 3\cdot 3\\ \\ {2}^{2}\cdot {3}^{2}\end{array}\)</div>
<p>Notice that the result is the same as we obtained with the factor tree method.</p>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Finding the prime factorization of a composite number using the ladder method.</h3>
<ol type="1">
<li>Divide the number by the smallest prime.</li>
<li>Continue dividing by that prime until it no longer divides evenly.</li>
<li>Divide by the next prime until it no longer divides evenly.</li>
<li>Continue until the quotient is a prime.</li>
<li>Write the composite number as the product of all the primes on the sides and top of the ladder.</li>
</ol>
</div>
</div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Find the prime factorization of \(120\) using the ladder method.</p>
</div>
<div>
<h3 data-type="title">Solution</h3>
<table data-label="" summary="The image shows multiple figures of ladder division. The first figure shows the division of 2 into 120 to get the quotient 60. Two is chosen to divide into 120 because it is the smallest prime number that is a factor of 120. This division is represented using a division bracket with 2 on the outside left of the bracket, 120 inside the bracket and 60 above the 120, outside the bracket. The next figure shows continuing to divide by 2 until it no longer divides evenly. The previous division repeated, but now with another division bracket written around the 60 with a 2 on the outside left of the bracket and a 30 above the 60, outside of the bracket. Since 30 is still divisible by 2 another division bracket is written around the 30 with a 2 on the outside left of the bracket and a 15 above the 30, outside of the bracket. The number 15 is not divisible by 2, but it is divisible by the next prime, 3. The next figure shows the previous figure with another division bracket written around the 15 with a 3 on the outside left of the bracket and a 5 above the 15, outside of the bracket. The quotient 5 is prime, so the ladder is complete.  The prime factorization of the number 120 is made up of all of the numbers from outside of the division brackets which are 2, 2, 2, 3, and 5. The prime factorization can be written as 2 times 2 times 2 times 3 times 5 or using exponents for repeated multiplication of 2 it can be written as 2 to the third power times 3 times 5.">
<tbody>
<tr>
<td>Divide the number by the smallest prime, which is 2.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/ddad7750-aef7-4f77-bf41-bc44ff2c97b4"/></td>
</tr>
<tr>
<td>Continue dividing by 2 until it no longer divides evenly.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/02bac9db-74c6-427f-aa08-082858b9e3d4"/></td>
</tr>
<tr>
<td>Divide by the next prime, 3.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/ec9a1810-2d12-4ab9-a06f-6e2f19240cfd"/></td>
</tr>
<tr>
<td>The quotient, 5, is prime, so the ladder is complete. Write the prime factorization of 120.</td>
<td>\(2\cdot 2\cdot 2\cdot 3\cdot 5\)

<div> </div>

\({2}^{3}\cdot 3\cdot 5\)</td>
</tr>
</tbody>
</table>
<p>Check this yourself by multiplying the factors. The result should be \(120.\)</p>
</div>
</div>
</div>
</div>
</div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Find the prime factorization of \(48\) using the ladder method.</p>
</div>
<div>
<h3 data-type="title">Solution</h3>
<table data-label="" summary="The image shows multiple figures of ladder division. The first figure shows the division of 2 into 48 to get the quotient 24. Two is chosen to divide into 48 because it is the smallest prime number that is a factor of 48. This division is represented using a division bracket with 2 on the outside left of the bracket, 48 inside the bracket and 24 above the 48, outside the bracket. The next figure shows continuing to divide by 2 until it no longer divides evenly. The previous division is repeated, but now with another division bracket written around the 24 with a 2 on the outside left of the bracket and a 12 above the 24, outside of the bracket. Since 12 is still divisible by 2 another division bracket is written around the 12 with a 2 on the outside left of the bracket and a 6 above the 30, outside of the bracket. The number 6 is still divisible by 2 so, another division bracket is written around the 6 with a 2 on the outside left of the bracket and a 3 above the 6, outside of the bracket. The quotient 3 is prime, so the ladder is complete.  The prime factorization of the number 48 is made up of all of the numbers from outside of the division brackets which are 2, 2, 2, 2, and 3. The prime factorization can be written as 2 times 2 times 2 times 2 times 3 or using exponents for repeated multiplication of 2 it can be written as 2 to the fourth power times 3.">
<tbody>
<tr>
<td>Divide the number by the smallest prime, 2.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/32bbf13c-43ff-4290-bb0a-1607083834dd"/></td>
</tr>
<tr>
<td>Continue dividing by 2 until it no longer divides evenly.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/ea2cb32d-34c2-4eea-ad55-740cc9aba3c0"/></td>
</tr>
<tr>
<td>The quotient, 3, is prime, so the ladder is complete. Write the prime factorization of 48.</td>
<td>\(2\cdot 2\cdot 2\cdot 2\cdot 3\)

<div> </div>

\({2}^{4}\cdot 3\)</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
</div>
<h2>Optional Videos by Mathispower4u</h2>
<h3>Example 1: Prime Factorization</h3>
<p><iframe allowfullscreen="allowfullscreen" frameborder="0" height="360" src="https://www.youtube.com/embed/2K5pBvb7Sss?rel=0&amp;modestbranding=1&amp;cc_lang_pref=en&amp;cc_load_policy=1" width="640"></iframe></p>
<h3>Example 2: Prime Factorization</h3>
<p><iframe allowfullscreen="allowfullscreen" frameborder="0" height="360" src="https://www.youtube.com/embed/zpVRADh86jU?rel=0&amp;modestbranding=1&amp;cc_lang_pref=en&amp;cc_load_policy=1" width="640"></iframe></p>
<h3>Example 3: Prime Factorization</h3>
<p><iframe allowfullscreen="allowfullscreen" frameborder="0" height="360" src="https://www.youtube.com/embed/uQi_O4D-SZg?rel=0&amp;modestbranding=1&amp;cc_lang_pref=en&amp;cc_load_policy=1" width="640"></iframe></p>
<h3>Example 1: Prime Factorization Using Stacked Division</h3>
<p><iframe allowfullscreen="allowfullscreen" frameborder="0" height="360" src="https://www.youtube.com/embed/V_wBWdndCuw?rel=0&amp;modestbranding=1&amp;cc_lang_pref=en&amp;cc_load_policy=1" width="640"></iframe></p>
<h3>Example 2: Prime Factorization Using Stacked Division</h3>
<p><iframe allowfullscreen="allowfullscreen" frameborder="0" height="360" src="https://www.youtube.com/embed/hQeJ36s1AAs?rel=0&amp;modestbranding=1&amp;cc_lang_pref=en&amp;cc_load_policy=1" width="640"><span class="mce_SELRES_start" data-mce-type="bookmark" style="display: inline-block; width: 0px; overflow: hidden; line-height: 0;">﻿</span></iframe></p>

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Prime Factorization of a Composite Number

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Prime Factorization of a Composite Number

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