<h2>Identifying Multiples of Numbers</h2>
<p>Annie is counting the shoes in her closet. The shoes are matched in pairs, so she doesn’t have to count each one. She counts by twos: \(2,4,6,8,10,12.\) She has \(12\) shoes in her closet.</p>
<p>The numbers \(2,4,6,8,10,12\) are called multiples of \(2.\) Multiples of \(2\) can be written as the product of a counting number and \(2.\) The first six multiples of \(2\) are given below.</p>
<div data-label="">\(\begin{array}{l}1\cdot 2=2\\ 2\cdot 2=4\\ 3\cdot 2=6\\ 4\cdot 2=8\\ 5\cdot 2=10\\ 6\cdot 2=12\end{array}\)</div>
<p>A <strong>multiple of a number</strong> is the product of the number and a counting number. So a multiple of \(3\) would be the product of a counting number and \(3.\) Below are the first six multiples of \(3.\)</p>
<div data-label="">\(\begin{array}{l}1\cdot 3=3\\ 2\cdot 3=6\\ 3\cdot 3=9\\ 4\cdot 3=12\\ 5\cdot 3=15\\ 6\cdot 3=18\end{array}\)</div>
<p>We can find the multiples of any number by continuing this process. The table below shows the multiples of \(2\) through \(9\) for the first twelve counting numbers.</p>
<table summary="The table has nine rows and thirteen columns. The first row is a header row. The first row, first column labels the first row as the counting numbers. The remaining columns are labeled 1 through 12. Starting in the second row, the first column list the categories: “Multiples of 2”,  “Multiples of 3”,  “Multiples of 4”, “Multiples of 5”,  “Multiples of 6”,  “Multiples of 7”, “Multiples of 8”, and  “Multiples of 9”. Under the “1” column are the values: 1, 2, 3, 4, 5, 6, 7, 8, and 9. Under the “2” column are the values: 4, 6, 8, 10, 12, 14, 16, and 18. Under the “3”column are the values: 3, 6, 9, 12, 15, 18, 24, and 27. Under the “4” column are the values: 8, 12, 16, 20, 24, 28, 32, and 36. Under the “5” column are the values: 10, 15, 20, 25, 30, 35, 40,  and 45. Under the “6” column are the values: 12, 18, 24, 30, 36, 42, 48, and 54. Under the “7” column are the values: 14, 21, 28, 35, 42, 49, 56, and 63. Under the “8” column are the values: 16, 24, 32, 40, 48, 56, 64, and 72. Under the “9” column are the values: 18, 27, 36, 45, 54, 63, 72, and 81. Under the “10” column are the values: 20, 30, 40, 50, 60, 70, 80, and 90. Under the “11” column are the values: 22, 33, 44, 55, 66, 77, 88, and 99. Under the “12” column are the values: 24, 36, 48, 60, 72, 84, 96, and 108.">
<thead>
<tr>
<th><strong>Counting Number</strong></th>
<th>\(1\)</th>
<th>\(2\)</th>
<th>\(3\)</th>
<th>\(4\)</th>
<th>\(5\)</th>
<th>\(6\)</th>
<th>\(7\)</th>
<th>\(8\)</th>
<th>\(9\)</th>
<th>\(10\)</th>
<th>\(11\)</th>
<th>\(12\)</th>
</tr>
</thead>
<tbody>
<tr>
<td>\(\text{Multiples of}\phantom{\rule{0.2em}{0ex}}2\)</td>
<td>\(2\)</td>
<td>\(4\)</td>
<td>\(6\)</td>
<td>\(8\)</td>
<td>\(10\)</td>
<td>\(12\)</td>
<td>\(14\)</td>
<td>\(16\)</td>
<td>\(18\)</td>
<td>\(20\)</td>
<td>\(22\)</td>
<td>\(24\)</td>
</tr>
<tr>
<td>\(\text{Multiples of}\phantom{\rule{0.2em}{0ex}}3\)</td>
<td>\(3\)</td>
<td>\(6\)</td>
<td>\(9\)</td>
<td>\(12\)</td>
<td>\(15\)</td>
<td>\(18\)</td>
<td>\(21\)</td>
<td>\(24\)</td>
<td>\(27\)</td>
<td>\(30\)</td>
<td>\(33\)</td>
<td>\(36\)</td>
</tr>
<tr>
<td>\(\text{Multiples of}\phantom{\rule{0.2em}{0ex}}4\)</td>
<td>\(4\)</td>
<td>\(8\)</td>
<td>\(12\)</td>
<td>\(16\)</td>
<td>\(20\)</td>
<td>\(24\)</td>
<td>\(28\)</td>
<td>\(32\)</td>
<td>\(36\)</td>
<td>\(40\)</td>
<td>\(44\)</td>
<td>\(48\)</td>
</tr>
<tr>
<td>\(\text{Multiples of}\phantom{\rule{0.2em}{0ex}}5\)</td>
<td>\(5\)</td>
<td>\(10\)</td>
<td>\(15\)</td>
<td>\(20\)</td>
<td>\(25\)</td>
<td>\(30\)</td>
<td>\(35\)</td>
<td>\(40\)</td>
<td>\(45\)</td>
<td>\(50\)</td>
<td>\(55\)</td>
<td>\(60\)</td>
</tr>
<tr>
<td>\(\text{Multiples of}\phantom{\rule{0.2em}{0ex}}6\)</td>
<td>\(6\)</td>
<td>\(12\)</td>
<td>\(18\)</td>
<td>\(24\)</td>
<td>\(30\)</td>
<td>\(36\)</td>
<td>\(42\)</td>
<td>\(48\)</td>
<td>\(54\)</td>
<td>\(60\)</td>
<td>\(66\)</td>
<td>\(72\)</td>
</tr>
<tr>
<td>\(\text{Multiples of}\phantom{\rule{0.2em}{0ex}}7\)</td>
<td>\(7\)</td>
<td>\(14\)</td>
<td>\(21\)</td>
<td>\(28\)</td>
<td>\(35\)</td>
<td>\(42\)</td>
<td>\(49\)</td>
<td>\(56\)</td>
<td>\(63\)</td>
<td>\(70\)</td>
<td>\(77\)</td>
<td>\(84\)</td>
</tr>
<tr>
<td>\(\text{Multiples of}\phantom{\rule{0.2em}{0ex}}8\)</td>
<td>\(8\)</td>
<td>\(16\)</td>
<td>\(24\)</td>
<td>\(32\)</td>
<td>\(40\)</td>
<td>\(48\)</td>
<td>\(56\)</td>
<td>\(64\)</td>
<td>\(72\)</td>
<td>\(80\)</td>
<td>\(88\)</td>
<td>\(96\)</td>
</tr>
<tr>
<td>\(\text{Multiples of}\phantom{\rule{0.2em}{0ex}}9\)</td>
<td>\(9\)</td>
<td>\(18\)</td>
<td>\(27\)</td>
<td>\(36\)</td>
<td>\(45\)</td>
<td>\(54\)</td>
<td>\(63\)</td>
<td>\(72\)</td>
<td>\(81\)</td>
<td>\(90\)</td>
<td>\(99\)</td>
<td>\(108\)</td>
</tr>
</tbody>
</table>
<div class="ui-has-child-title" data-type="note">
<header>
<div data-type="title">
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Multiple of a Number</h3>
<p>A number is a <strong>multiple</strong> of \(n\) if it is the product of a counting number and \(n.\)</p>
</div>
</div>
</div>
</header>
</div>
<p>Recognizing the patterns for multiples of \(2,5,10,\text{and}\phantom{\rule{0.2em}{0ex}}3\) will be helpful to you as you continue in this tutorial.</p>
<div class="ns-tutorial">
<p>The figure below shows the counting numbers from \(1\) to \(50.\) Multiples of \(2\) are highlighted. Do you notice a pattern?</p>
<div class="wp-caption alignnone" style="width: 526px"><img alt="The image shows a chart with five rows and ten columns. The first row lists the numbers from 1 to 10. The second row lists the numbers from 11 to 20. The third row lists the numbers from 21 to 30. The fourth row lists the numbers from 31 and 40. The fifth row lists the numbers from 41 to 50. All factors of 2 are highlighted in blue." height="96" src="https://data.ulearngo.com/assets/dbc06eda-4317-46b5-8f7d-ff03f200e440" width="526"/><p class="wp-caption-text">Multiples of \(2\) between \(1\) and \(50\)</p></div>
<p>The last digit of each highlighted number in the figure below is either \(0,2,4,6,\text{or}\phantom{\rule{0.2em}{0ex}}8.\) This is true for the product of \(2\) and any counting number. So, to tell if any number is a multiple of \(2\) look at the last digit. If it is \(0,2,4,6,\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}8,\) then the number is a multiple of \(2.\)</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<p>Determine whether each of the following is a multiple of \(2\text{:}\)</p>
<ol type="1">
<li>\(\phantom{\rule{0.2em}{0ex}}489\phantom{\rule{0.2em}{0ex}}\)</li>
<li>\(\phantom{\rule{0.2em}{0ex}}3,714\)</li>
</ol>
</div>
<div>
<h3 data-type="title">Solution</h3>
<table data-label="" summary=".">
<tbody>
<tr>
<td>1. </td>
<td> </td>
</tr>
<tr>
<td>Is 489 a multiple of 2?</td>
<td> </td>
</tr>
<tr>
<td>Is the last digit 0, 2, 4, 6, or 8?</td>
<td>No.</td>
</tr>
<tr>
<td> </td>
<td>489 is not a multiple of 2.</td>
</tr>
</tbody>
</table>
<table data-label="" summary=".">
<tbody>
<tr>
<td>2.</td>
<td> </td>
</tr>
<tr>
<td>Is 3,714 a multiple of 2?</td>
<td> </td>
</tr>
<tr>
<td>Is the last digit 0, 2, 4, 6, or 8?</td>
<td>Yes.</td>
</tr>
<tr>
<td> </td>
<td>3,714 is a multiple of 2.</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
</div>
<p>Now let’s look at multiples of \(5.\) The figure below highlights all of the multiples of \(5\) between \(1\) and \(50.\) What do you notice about the multiples of \(5?\)</p>
<div class="wp-caption alignnone" style="width: 526px"><img alt="The image shows a chart with five rows and ten columns. The first row lists the numbers from 1 to 10. The second row lists the numbers from 11 to 20. The third row lists the numbers from 21 to 30. The fourth row lists the numbers from 31 and 40. The fifth row lists the numbers from 41 to 50. All factors of 5 are highlighted in blue." height="96" src="https://data.ulearngo.com/assets/ac5c68af-ba8b-460a-8302-316cb3b5439c" width="526"/><p class="wp-caption-text">Multiples of \(5\) between \(1\) and \(50\)</p></div>
<p>All multiples of \(5\) end with either \(5\) or \(0.\) Just like we identify multiples of \(2\) by looking at the last digit, we can identify multiples of \(5\) by looking at the last digit.</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<p>Determine whether each of the following is a multiple of \(5\text{:}\phantom{\rule{0.2em}{0ex}}\)</p>
<ol type="1">
<li>579</li>
<li>880</li>
</ol>
</div>
<div>
<h3 data-type="title">Solution</h3>
<table data-label="" summary=".">
<tbody>
<tr>
<td>1.</td>
<td> </td>
</tr>
<tr>
<td>Is 579 a multiple of 5?</td>
<td> </td>
</tr>
<tr>
<td>Is the last digit 5 or 0?</td>
<td>No.</td>
</tr>
<tr>
<td> </td>
<td>579 is not a multiple of 5.</td>
</tr>
</tbody>
</table>
<table data-label="" summary=".">
<tbody>
<tr>
<td>2.</td>
<td> </td>
</tr>
<tr>
<td>Is 880 a multiple of 5?</td>
<td> </td>
</tr>
<tr>
<td>Is the last digit 5 or 0?</td>
<td>Yes.</td>
</tr>
<tr>
<td> </td>
<td>880 is a multiple of 5.</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
<p>The figure below highlights the multiples of \(10\) between \(1\) and \(50.\) All multiples of \(10\) all end with a zero.</p>
<div class="wp-caption alignnone" style="width: 526px"><img alt="The image shows a chart with five rows and ten columns. The first row lists the numbers from 1 to 10. The second row lists the numbers from 11 to 20. The third row lists the numbers from 21 to 30. The fourth row lists the numbers from 31 and 40. The fifth row lists the numbers from 41 to 50. All factors of 10 are highlighted in blue." height="96" src="https://data.ulearngo.com/assets/60030278-ab6c-4aa5-985e-02e1fda38b4c" width="526"/><p class="wp-caption-text">Multiples of \(10\) between \(1\) and \(50\)</p></div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<p>Determine whether each of the following is a multiple of \(10\text{:}\phantom{\rule{0.2em}{0ex}}\)</p>
<ol type="1">
<li>\(\phantom{\rule{0.2em}{0ex}}425\phantom{\rule{0.2em}{0ex}}\)</li>
<li>\(\phantom{\rule{0.2em}{0ex}}350\)</li>
</ol>
</div>
<div>
<h3 data-type="title">Solution</h3>
<table data-label="" summary=".">
<tbody>
<tr>
<td>1.</td>
<td> </td>
</tr>
<tr>
<td>Is 425 a multiple of 10?</td>
<td> </td>
</tr>
<tr>
<td>Is the last digit zero?</td>
<td>No.</td>
</tr>
<tr>
<td> </td>
<td>425 is not a multiple of 10.</td>
</tr>
</tbody>
</table>
<table data-label="" summary=".">
<tbody>
<tr>
<td>2.</td>
<td> </td>
</tr>
<tr>
<td>Is 350 a multiple of 10?</td>
<td> </td>
</tr>
<tr>
<td>Is the last digit zero?</td>
<td>Yes.</td>
</tr>
<tr>
<td> </td>
<td>350 is a multiple of 10.</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
<p>The figure below highlights multiples of \(3.\) The pattern for multiples of \(3\) is not as obvious as the patterns for multiples of \(2,5,\text{and}\phantom{\rule{0.2em}{0ex}}10.\)</p>
<div class="wp-caption alignnone" style="width: 526px"><img alt="The image shows a chart with five rows and ten columns. The first row lists the numbers from 1 to 10. The second row lists the numbers from 11 to 20. The third row lists the numbers from 21 to 30. The fourth row lists the numbers from 31 and 40. The fifth row lists the numbers from 41 to 50. All factors of 3 are highlighted in blue." height="96" src="https://data.ulearngo.com/assets/05341bf5-8baf-4b24-b892-685ac4735716" width="526"/><p class="wp-caption-text">Multiples of \(3\) between \(1\) and \(50\)</p></div>
<p>Unlike the other patterns we’ve examined so far, this pattern does not involve the last digit. The pattern for multiples of \(3\) is based on the sum of the digits. If the sum of the digits of a number is a multiple of \(3,\) then the number itself is a multiple of \(3.\) See the table.</p>
<table summary="This table has two rows and nine columns. The first column labels the two rows. The first row is “Multiple of 3” and the second row is “Sum of digits.” The second column shows 3 in the first row and 3 in the second row. The third column shows 6 in the first row and 6 in the second row. The fourth column shows 9 in the first row and 9 in the second row. The fifth column shows 12 in the first row and 1 plus 2 is 3 in the second row. The sixth column shows 15 in the first row and 1 plus 5 is 6 in the second row. The seventh column shows 18 in the first row and 1 plus 8 is 9 in the second row. The eighth column shows 21 in the first row and 2 plus 1 is 3 in the second row. The last column shows 24 in the first row and 2 plus 4 is 6 in the second row.">
<tbody>
<tr>
<td>\(\mathbf{\text{Multiple of 3}}\)</td>
<td>\(3\)</td>
<td>\(6\)</td>
<td>\(9\)</td>
<td>\(12\)</td>
<td>\(15\)</td>
<td>\(18\)</td>
<td>\(21\)</td>
<td>\(24\)</td>
</tr>
<tr>
<td>\(\mathbf{\text{Sum of digits}}\)</td>
<td>\(3\)</td>
<td>\(6\)</td>
<td>\(9\)</td>
<td>\(\begin{array}{c}\hfill 1+2\hfill \\ \hfill 3\hfill \end{array}\)</td>
<td>\(\begin{array}{c}\hfill 1+5\hfill \\ \hfill 6\hfill \end{array}\)</td>
<td>\(\begin{array}{c}\hfill 1+8\hfill \\ \hfill 9\hfill \end{array}\)</td>
<td>\(\begin{array}{c}\hfill 2+1\hfill \\ \hfill 3\hfill \end{array}\)</td>
<td>\(\begin{array}{c}\hfill 2+4\hfill \\ \hfill 6\hfill \end{array}\)</td>
</tr>
</tbody>
</table>
<p>Consider the number \(42.\) The digits are \(4\) and \(2,\) and their sum is \(4+2=6.\) Since \(6\) is a multiple of \(3,\) we know that \(42\) is also a multiple of \(3.\)</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<div>
<p>Determine whether each of the given numbers is a multiple of \(3\text{:}\phantom{\rule{0.2em}{0ex}}\)</p>
<ol type="1">
<li>645</li>
<li>\(\phantom{\rule{0.2em}{0ex}}10,519\)</li>
</ol>
</div>
<div>
<h3 data-type="title">Solution</h3>
<p>Is \(645\) a multiple of \(3?\)</p>
<table data-label="" summary=".">
<tbody>
<tr>
<td>Find the sum of the digits.</td>
<td>\(6+4+5=15\)</td>
</tr>
<tr>
<td>Is 15 a multiple of 3?</td>
<td>Yes.</td>
</tr>
<tr>
<td>If we're not sure, we could add its digits to find out. We can check it by dividing 645 by 3.</td>
<td>\(645÷3\)</td>
</tr>
<tr>
<td>The quotient is 215.</td>
<td>\(3\cdot 215=645\)</td>
</tr>
</tbody>
</table>
<p>Is \(10,519\) a multiple of \(3?\)</p>
<table data-label="" summary=".">
<tbody>
<tr>
<td>Find the sum of the digits.</td>
<td>\(1+0+5+1+9=16\)</td>
</tr>
<tr>
<td>Is 16 a multiple of 3?</td>
<td>No.</td>
</tr>
<tr>
<td>So 10,519 is not a multiple of 3 either..</td>
<td>\(645÷3\)</td>
</tr>
<tr>
<td>We can check this by dividing by 10,519 by 3.</td>
<td>\(\begin{array}{c}3,506\text{R}1\\ \hfill 3\overline{)10,519}\phantom{\rule{1em}{0ex}}\end{array}\)</td>
</tr>
</tbody>
</table>
<p>When we divide \(10,519\) by \(3,\) we do not get a counting number, so \(10,519\) is not the product of a counting number and \(3.\) It is not a multiple of \(3.\)</p>
</div>
</div>
</div>
</div>
</div>
<p>Look back at the charts where you highlighted the multiples of \(2,\) of \(5,\) and of \(10.\) Notice that the multiples of \(10\) are the numbers that are multiples of both \(2\) and \(5.\) That is because \(10=2\cdot 5.\) Likewise, since \(6=2\cdot 3,\) the multiples of \(6\) are the numbers that are multiples of both \(2\) and \(3.\)</p>

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Identifying Multiples of Numbers

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Identifying Multiples of Numbers

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