<p>Theorems,calculus,quadratic,equations ,geometry, surd,logarithms and longitudes and latitudes .
Please explain more of it</p>


<h2>Finding the Least Common Multiple (LCM) of Two Numbers</h2>
<p>One of the reasons we look at multiples and primes is to use these techniques to find the least common multiple of two numbers. This will be useful when we add and subtract fractions with different denominators.</p>
<h3 data-type="title">Listing Multiples Method</h3>
<p>A common multiple of two numbers is a number that is a multiple of both numbers. Suppose we want to find common multiples of \(10\) and \(25.\) We can list the first several multiples of each number. Then we look for multiples that are common to both lists—these are the common multiples.</p>
<p>\(10:10,20,30,40,\mathbf{50},60,70,80,90,\mathbf{100},110,…\)</p>
<p>\(25:25,\mathbf{50},75,\mathbf{100},125,…\)</p>
<p>We see that \(50\) and \(100\) appear in both lists. They are common multiples of \(10\) and \(25.\) We would find more common multiples if we continued the list of multiples for each.</p>
<p>The smallest number that is a multiple of two numbers is called the <strong>least common multiple</strong> (LCM). So the least LCM of \(10\) and \(25\) is \(50.\)</p>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Find the least common multiple (LCM) of two numbers by listing multiples.</h3>
<ol type="1">
<li>List the first several multiples of each number.</li>
<li>Look for multiples common to both lists. If there are no common multiples in the lists, write out additional multiples for each number.</li>
<li>Look for the smallest number that is common to both lists.</li>
<li>This number is the LCM.</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 LCM of \(15\) and \(20\) by listing multiples. </p>
</div>
<div>
<h3 data-type="title">Solution</h3>
<p>List the first several multiples of \(15\) and of \(20.\) Identify the first common multiple.</p>
<p>\(\begin{array}{l}\text{15:}\phantom{\rule{0.2em}{0ex}}15,30,45,\phantom{\rule{0.2em}{0ex}}60,75,90,105,120\hfill \\ \text{20:}\phantom{\rule{0.2em}{0ex}}20,40,\phantom{\rule{0.2em}{0ex}}60,80,100,120,140,160\hfill \end{array}\)</p>
<p>The smallest number to appear on both lists is \(60,\) so \(60\) is the least common multiple of \(15\) and \(20.\)</p>
<p>Notice that \(120\) is on both lists, too. It is a common multiple, but it is not the least common multiple.</p>
</div>
</div>
</div>
</div>
</div>
<h3 data-type="title">Prime Factors Method</h3>
<p>Another way to find the least common multiple of two numbers is to use their prime factors. We’ll use this method to find the LCM of \(12\) and \(18.\)</p>
<p>We start by finding the prime factorization of each number.</p>
<div data-label="">\(12=2\cdot 2\cdot 3\phantom{\rule{3em}{0ex}}18=2\cdot 3\cdot 3\)</div>
<p>Then we write each number as a product of primes, matching primes vertically when possible.</p>
<div data-label="">\(\begin{array}{}12=2\cdot 2\cdot 3\phantom{\rule{3em}{0ex}} \\ 18=2\cdot \: \: \: \: \; 3\cdot 3\end{array}\)</div>
<p>Now we bring down the primes in each column. The LCM is the product of these factors.</p>
<p><img alt="The image shows the prime factorization of 12 written as the equation 12 equals 2 times 2 times 3. Below this equation is another showing the prime factorization of 18 written as the equation 18 equals 2 times 3 times 3. The two equations line up vertically at the equal symbol. The first 2 in the prime factorization of 12 aligns with the 2 in the prime factorization of 18. Under the second 2 in the prime factorization of 12 is a gap in the prime factorization of 18. Under the 3 in the prime factorization of 12 is the first 3 in the prime factorization of 18. The second 3 in the prime factorization has no factors above it from the prime factorization of 12. A horizontal line is drawn under the prime factorization of 18. Below this line is the equation LCM equal to 2 times 2 times 3 times 3. Arrows are drawn down vertically from the prime factorization of 12 through the prime factorization of 18 ending at the LCM equation. The first arrow starts at the first 2 in the prime factorization of 12 and continues down through the 2 in the prime factorization of 18. Ending with the first 2 in the LCM. The second arrow starts at the next 2 in the prime factorization of 12 and continues down through the gap in the prime factorization of 18. Ending with the second 2 in the LCM. The third arrow starts at the 3 in the prime factorization of 12 and continues down through the first 3 in the prime factorization of 18. Ending with the first 3 in the LCM. The last arrow starts at the second 3 in the prime factorization of 18 and points down to the second 3 in the LCM." src="https://data.ulearngo.com/assets/5d7884cd-2a38-46ae-9c35-3916aa9a7704"/></p>
<p>Notice that the prime factors of \(12\) and the prime factors of \(18\) are included in the LCM. By matching up the common primes, each common prime factor is used only once. This ensures that \(36\) is the least common multiple.</p>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Finding the LCM using the prime factors method.</h3>
<ol type="1">
<li>Find the prime factorization of each number.</li>
<li>Write each number as a product of primes, matching primes vertically when possible.</li>
<li>Bring down the primes in each column.</li>
<li>Multiply the factors to get the LCM.</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 LCM of \(15\) and \(18\) using the prime factors method.</p>
</div>
<div>
<h3 data-type="title">Solution</h3>
<table data-label="" summary="In the figure the prime factorization of 15 is written as the equation 15 equals 3 times 5. Next to that is the prime factorization of 18 written as the equation 2 times 3 times 3. Below that is the prime factorization equations of 15 and 18 written to align vertically with the equation for 15 above the equation for 18. These equations are written so that similar prime factors line up vertically. Below this is the prime factorization equations aligned vertically again with a horizontal line drawn under the prime factorization of 18. Below this line is the equation LCM equal to 2 times 3 times 3 times 5. Arrows are drawn down vertically from the prime factorization of 15 through the prime factorization of 18 ending at the LCM equation. Since there is no 2 in the prime factorization of 15, the first arrow starts at the 2 in the prime factorization of 18 and points down to the 2 in the LCM. The second arrow starts at the first 3 in the prime factorization of 15 and continues down through the first 3 in the prime factorization of 18. Ending with the first 3 in the LCM. Since there are no more three's in the prime factorization of 15, the next arrow starts at the second 3 in the prime factorization of 18 and points down to the second 3 in the LCM. Since there are no more factors in the prime factorization of 18, the last arrow starts at the 5 in the prime factorization of 15 and points down to through the empty space at the end of the prime factorization of 18 to the 5 in the LCM. The least common multiple of 15 and 18 is 2 times 3 times 3 times 5 which is 90.">
<tbody>
<tr>
<td>Write each number as a product of primes.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/64558887-5bd7-4ffc-a000-386475ee0655"/></td>
</tr>
<tr>
<td>Write each number as a product of primes, matching primes vertically when possible.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/b816d847-6a9b-4785-a7da-1e646aa25e78"/></td>
</tr>
<tr>
<td>Bring down the primes in each column.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/20317e61-09a0-48c1-a7eb-76277e001477"/></td>
</tr>
<tr>
<td>Multiply the factors to get the LCM.</td>
<td>\(\text{LCM}=2\cdot 3\cdot 3\cdot 5\)

<div> </div>

The LCM of 15 and 18 is 90.</td>
</tr>
</tbody>
</table>
</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 LCM of \(50\) and \(100\) using the prime factors method.</p>
</div>
<div>
<h3 data-type="title">Solution</h3>
<table data-label="" summary="In the figure the prime factorization of 50 is written as the equation 50 equals 2 times 5 times 5. Next to that is the prime factorization of 100 written as the equation 2 times 2 times 5 times 5. Below that is the prime factorization equations of 50 and 100 written to align vertically with the equation for 50 above the equation for 100. These equations are written so that similar prime factors line up vertically. Below this is the prime factorization equations aligned vertically again with a horizontal line drawn under the prime factorization of 100. Below this line is the equation LCM equal to 2 times 2 times 5 times 5. Arrows are drawn down vertically from the prime factorization of 50 through the prime factorization of 100 ending at the LCM equation. Since there is only one 2 in the prime factorization of 50, the first arrow starts at the first 2 in the prime factorization of 100 and points down to the first 2 in the LCM. The second arrow starts at the first 2 in the prime factorization of 50 and continues down through the second 2 in the prime factorization of 100. Ending with the second 2 in the LCM. The next arrow starts at the first 5 in the prime factorization of 50 and continues through the first 5 in the prime factorization of 100. Ending in the first 5 in the LCM. The last arrow starts at the second 5 in the prime factorization of 50 and continues through the second 5 in the prime factorization of 100. Ending in the second 5 in the LCM. The least common multiple of 50 and 100 is 2 times 2 times 5 times 5 which is 100.">
<tbody>
<tr>
<td>Write the prime factorization of each number.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/7c524dd0-cc50-41c4-a7d2-66da26b9b568"/></td>
</tr>
<tr>
<td>Write each number as a product of primes, matching primes vertically when possible.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/666236b3-a21e-4371-9a97-3740806f6957"/></td>
</tr>
<tr>
<td>Bring down the primes in each column.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/abd8fdf8-b7ec-4376-84f3-8a1d98933106"/></td>
</tr>
<tr>
<td>Multiply the factors to get the LCM.</td>
<td>\(\text{LCM}=2\cdot 2\cdot 5\cdot 5\)

<div> </div>

The LCM of 50 and 100 is 100.</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
</div>
<div class="media-2 ui-has-child-title" data-type="note">
<header>
<div data-type="title">
<h2>Optional Videos by Mathispower4u</h2>
</div>
</header>
<h3>The Least Common Multiple</h3>
<iframe allowfullscreen="allowfullscreen" frameborder="0" height="360" src="https://www.youtube.com/embed/lPdrA-_I4IM?rel=0&amp;modestbranding=1&amp;cc_lang_pref=en&amp;cc_load_policy=1" width="640"></iframe>
<h3>Example: Determining the Least Common Multiple Using a List of Multiples</h3>
<iframe allowfullscreen="allowfullscreen" frameborder="0" height="360" src="https://www.youtube.com/embed/7twRSmgcrLM?rel=0&amp;modestbranding=1&amp;cc_lang_pref=en&amp;cc_load_policy=1" width="640"></iframe>
<h3>Example: Determining the Least Common Multiple Using Prime Factorization</h3>
<iframe allowfullscreen="allowfullscreen" frameborder="0" height="360" src="https://www.youtube.com/embed/hZvRDG-HgMY?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></div>

77e14597-4d58-4399-be3f-b1bb5c28bc54

Least Common Multiple (LCM) of Two Numbers

Mathematics studies measurement, relationships, and properties of quantities and sets, using numbers and symbols. Arithmetic, algebra, geometry, and calculus are popular topics in mathematics.

Mathematics

Learn to identify and simplify algebraic expressions, evaluate expressions, and translate word problems into algebraic equations in this comprehensive tutorial on the language of algebra.


Least Common Multiple (LCM) of Two Numbers

Lesson video

Lesson video

Lesson video

Track Your Learning Progress

Earlier discussion