<h2>Divide Whole Numbers</h2>
<p>We said that addition and subtraction are inverse operations because one undoes the other. Similarly, division is the inverse operation of multiplication. We know $12÷4=3$ because $3·4=12.$ Knowing all the multiplication number facts is very important when doing division.</p>
<p>We check our answer to division by multiplying the quotient by the divisor to determine if it equals the dividend. From the example in the previous lesson, we know $24÷8=3$ is correct because $3·8=24.$</p>
<div class="ns-school-emp">
<h3>Example</h3>
<div class="ns-school-defn">
<div data-type="example">
<section></section>
<section>
<div>
<p>Divide. Then check by multiplying. (a) $42÷6$ (b) $\frac{72}{9}$ (c) $7\overline{)63}$</p>
</div>
<div>
<section>
<h4>Solution</h4>
<table data-label="" summary=" ">
<tbody>
<tr>
<td>(a)</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td>$42÷6$</td>
</tr>
<tr>
<td>Divide 42 by 6.</td>
<td>$7$</td>
</tr>
<tr>
<td>Check by multiplying. $7·6$</td>
<td> </td>
</tr>
<tr>
<td>$42✓$</td>
<td> </td>
</tr>
</tbody>
</table>
<table data-label="" summary=" ">
<tbody>
<tr>
<td>(b)</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td>$\cfrac{72}{9}$</td>
</tr>
<tr>
<td>Divide 72 by 9.</td>
<td>$8$</td>
</tr>
<tr>
<td>Check by multiplying. $8·9$</td>
<td> </td>
</tr>
<tr>
<td>$72✓$</td>
<td> </td>
</tr>
</tbody>
</table>
<table data-label="" summary="a">
<tbody>
<tr>
<td>(c)</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td>$7\overline{)63}$</td>
</tr>
<tr>
<td>Divide 63 by 7.</td>
<td>$9$</td>
</tr>
<tr>
<td>Check by multiplying.<br/>
 $9·7$</td>
<td> </td>
</tr>
<tr>
<td>$63✓$</td>
<td> </td>
</tr>
</tbody>
</table>
</section>
</div>
</section>
</div>
</div>
</div>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Extra:</h3>
<div data-type="note">
<section></section>
<section>
<div>
<p>Divide. Then check by multiplying:</p>
<p>(a) $54÷6$ (b) $\frac{27}{9}$</p>
</div>
<div>
<h4>Answer</h4>
<section>
<p>(a) 9 (b) 3</p>
</section>
</div>
</section>
</div>
</div>
</div>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Extra:</h3>
<div data-type="note">
<section>
<div>
<section>
<div>
<p>Divide. Then check by multiplying:</p>
<p>(a) $\frac{36}{9}$ (b) $8\overline{)40}$</p>
</div>
<div>
<h4>Solution</h4>
<section>
<p>(a) 4 (b) 5</p>
</section>
</div>
</section>
</div>
</section>
</div>
</div>
</div>
<p>What is the quotient when you divide a number by itself?</p>
<div data-label="">$\cfrac{15}{15}=1\phantom{\rule{0.2em}{0ex}}\mathrm{because}\phantom{\rule{0.2em}{0ex}}1·15=15$</div>
<p>Dividing any number $\mathrm{(except 0)}$ by itself produces a quotient of $1.$ Also, any number divided by $1$ produces a quotient of the number. These two ideas are stated in the Division Properties of One.</p>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Division Properties of One:</h3>
<section>
<table data-label="" summary="a">
<tbody>
<tr>
<td>Any number (except 0) divided by itself is one.</td>
<td>$a÷a=1$</td>
</tr>
<tr>
<td>Any number divided by one is the same number.</td>
<td>$a÷1=a$</td>
</tr>
</tbody>
</table>
</section>
</div>
</div>
<div class="ns-school-emp">
<h3>Example</h3>
<div class="ns-school-defn">
<div data-type="example">
<section></section>
<section>
<div>
<p>Divide. Then check by multiplying:</p>
<ol style="list-style-type: lower-alpha;" type="1">
<li>$11÷11$</li>
<li>$\frac{19}{1}$</li>
<li>$1\overline{)7}$</li>
</ol>
</div>
<div>
<section>
<h4>Solution</h4>
<table data-label="" summary="t">
<tbody>
<tr>
<td>a.</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td>$11÷11$</td>
</tr>
<tr>
<td>A number divided by itself is 1.</td>
<td>$1$</td>
</tr>
<tr>
<td>Check by multiplying.<br/>
 $1·11$</td>
<td> </td>
</tr>
<tr>
<td>$11✓$</td>
<td> </td>
</tr>
</tbody>
</table>
<table data-label="" summary="t">
<tbody>
<tr>
<td>b.</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td>$\frac{19}{1}$</td>
</tr>
<tr>
<td>A number divided by 1 equals itself.</td>
<td>$19$</td>
</tr>
<tr>
<td>Check by multiplying.<br/>
 $19·1$</td>
<td> </td>
</tr>
<tr>
<td>$19✓$</td>
<td> </td>
</tr>
</tbody>
</table>
<table data-label="" summary="t">
<tbody>
<tr>
<td>c.</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td>$1\overline{)7}$</td>
</tr>
<tr>
<td>A number divided by 1 equals itself.</td>
<td>$7$</td>
</tr>
<tr>
<td>Check by multiplying. $7·1$</td>
<td> </td>
</tr>
<tr>
<td>$7✓$</td>
<td> </td>
</tr>
</tbody>
</table>
</section>
</div>
</section>
</div>
</div>
</div>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Extra:</h3>
<div data-type="note">
<section></section>
<section>
<div>
<p>Divide. Then check by multiplying:</p>
<p>(a) $14÷14$ (b) $\frac{27}{1}$</p>
</div>
<div>
<h4>Answer</h4>
<section>
<p>(a) 1 (b) 27</p>
</section>
</div>
</section>
</div>
</div>
</div>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Extra:</h3>
<div data-type="note">
<section>
<div>
<section>
<div>
<p>Divide. Then check by multiplying:</p>
<p>(a) $\frac{16}{1}$ (b) $1\overline{)4}$</p>
</div>
<div>
<h4>Solution</h4>
<section>
<p>(a) 16 (b) 4</p>
</section>
</div>
</section>
</div>
</section>
</div>
</div>
</div>
<p>Suppose we have $\mathrm{$0},$ and want to divide it among $3$ people. How much would each person get? Each person would get $\mathrm{$0}.$ Zero divided by any number is $0.$</p>
<p>Now suppose that we want to divide $\mathrm{$1}0$ by $0.$ That means we would want to find a number that we multiply by $0$ to get $10.$ This cannot happen because $0$ times any number is $0.$ Division by zero is said to be <em>undefined</em>.</p>
<p>These two ideas make up the Division Properties of Zero.</p>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Division Properties of Zero:</h3>
<section>
<table data-label="" summary="a">
<tbody>
<tr>
<td>Zero divided by any number is 0.</td>
<td>$0÷a=0$</td>
</tr>
<tr>
<td>Dividing a number by zero is undefined.</td>
<td>$a÷0$ undefined</td>
</tr>
</tbody>
</table>
</section>
</div>
</div>
<p>Another way to explain why division by zero is undefined is to remember that division is really repeated subtraction. How many times can we take away $0$ from $10?$ Because subtracting $0$ will never change the total, we will never get an answer. So we cannot divide a number by $0.$</p>
<div class="ns-school-emp">
<h3>Example</h3>
<div class="ns-school-defn">
<div data-type="example">
<section></section>
<section>
<div>
<p>Divide. Check by multiplying: (a) $0÷3$ (b) $10/0.$</p>
</div>
<div>
<section>
<h4>Solution</h4>
<table data-label="" summary=" ">
<tbody>
<tr>
<td>(a)</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td>$0÷3$</td>
</tr>
<tr>
<td>Zero divided by any number is zero.</td>
<td>$0$</td>
</tr>
<tr>
<td>Check by multiplying.

<div> </div>

$0·3$</td>
<td> </td>
</tr>
<tr>
<td>$0✓$</td>
<td> </td>
</tr>
</tbody>
</table>
<table data-label="" summary=" ">
<tbody>
<tr>
<td>(b)</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td>$10/0$</td>
</tr>
<tr>
<td>Division by zero is undefined.</td>
<td>undefined</td>
</tr>
</tbody>
</table>
</section>
</div>
</section>
</div>
</div>
</div>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Extra:</h3>
<div data-type="note">
<section>
<div>
<section>
<div>
<p>Divide. Then check by multiplying:</p>
<p>(a) $0÷2$ (b) $17/0$</p>
</div>
<div>
<h4>Solution</h4>
<section>
<p>(a) 0 (b) undefined</p>
</section>
</div>
</section>
</div>
</section>
</div>
</div>
</div>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Extra:</h3>
<div data-type="note">
<section>
<div>
<section>
<div>
<p>Divide. Then check by multiplying:</p>
<p>(a) $0÷6$ (b) $13/0$</p>
</div>
<div>
<h4>Solution</h4>
<section>
<p>(a) 0 (b) undefined</p>
</section>
</div>
</section>
</div>
</section>
</div>
</div>
</div>
<p>When the divisor or the dividend has more than one digit, it is usually easier to use the $4\overline{)12}$ notation. This process is called long division. Let’s work through the process by dividing $78$ by $3.$</p>
<table data-label="" style="width: 821px;" summary="This image has 2 columns. the left column contains instructions and the right column contains expressions. The exercises being worked out is 78 divided by 3. The first line reads: Divide the first digit of dividend 7, by the divisor, 3. The next line reads: the divisor 3 can go into 7 two times since 2 times 3 equals 6. Write the 2 above the 7 in the quotient. Next to this shows the expression 3 divided by 78, with the two above the seven in the quotient. The next line reads “multiply the 2 in the quotient by 3 and write the product, 6 under the 7”. The expression now shows the 6 under the seven. the next line reads “subtract that product from the first digit in the dividend. Subtract 7 minus 6. Write the difference,1, under the first digit in the dividend”. The expression now shows the 6 subtracted from the 7 and the difference of 1. The next line says to bring sown the next digit of the dividend. Bring down the 8. The expression now shows the 8 brought down next to the 1. In the next line, it says to divide 18 by the divisor, 3. The divisor 3 goes into 18 six times. Write 6 in the quotient above the 8. The expression shows the quotient 6 above the 8. Finally, it says to multiply the 6 in the quotient by the divisor and write the product, 18 under the dividend. Subtract 18 from 18. The expression shows 18 minus 18 in long division which equals zero.">
<tbody>
<tr>
<td style="width: 747px;">Divide the first digit of the dividend, 7, by the divisor, 3.</td>
<td style="width: 51px;"> </td>
</tr>
<tr>
<td style="width: 747px;">The divisor 3 can go into 7 two times since $2×3=6$. Write the 2 above the 7 in the quotient.</td>
<td style="width: 51px;"><img alt="CNX_BMath_Figure_01_05_043_img-02.png" src="https://data.ulearngo.com/assets/6583c325-0a1b-474e-96e9-ffb771273167"/></td>
</tr>
<tr>
<td style="width: 747px;">Multiply the 2 in the quotient by 2 and write the product, 6, under the 7.</td>
<td style="width: 51px;"><img alt="CNX_BMath_Figure_01_05_043_img-03.png" src="https://data.ulearngo.com/assets/bd8bf823-3a64-45ad-a221-fafb756d05f0"/></td>
</tr>
<tr>
<td style="width: 747px;">Subtract that product from the first digit in the dividend. Subtract $7-6$. Write the difference, 1, under the first digit in the dividend.</td>
<td style="width: 51px;"><img alt="CNX_BMath_Figure_01_05_043_img-04.png" src="https://data.ulearngo.com/assets/3ff54713-31d1-47ec-850f-d609e2a93b18"/></td>
</tr>
<tr>
<td style="width: 747px;">Bring down the next digit of the dividend. Bring down the 8.</td>
<td style="width: 51px;"><img alt="CNX_BMath_Figure_01_05_043_img-05.png" src="https://data.ulearngo.com/assets/141cbd18-75f3-4ddd-8cf4-03545768a986"/></td>
</tr>
<tr>
<td style="width: 747px;">Divide 18 by the divisor, 3. The divisor 3 goes into 18 six times.</td>
<td rowspan="2" style="width: 51px;"><img alt="CNX_BMath_Figure_01_05_043_img-06.png" src="https://data.ulearngo.com/assets/bd4adb4d-0385-4789-9727-c5abf5db5658"/></td>
</tr>
<tr>
<td data-valign="bottom" style="width: 747px;">Write 6 in the quotient above the 8.</td>
</tr>
<tr>
<td style="width: 747px;">Multiply the 6 in the quotient by the divisor and write the product, 18, under the dividend. Subtract 18 from 18.</td>
<td style="width: 51px;"><img alt="CNX_BMath_Figure_01_05_043_img-07.png" src="https://data.ulearngo.com/assets/93570ddd-c108-4e35-a3d7-5def753125f1"/></td>
</tr>
</tbody>
</table>
<p>We would repeat the process until there are no more digits in the dividend to bring down. In this problem, there are no more digits to bring down, so the division is finished.</p>
<div data-label="">So $\phantom{\rule{0.2em}{0ex}}78÷3=26.$</div>
<p>Check by multiplying the quotient times the divisor to get the dividend. Multiply $26\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}3$ to make sure that product equals the dividend, $78.$</p>
<div>$\begin{array}{r}\stackrel{1}{2}6 &amp; \\ ×\;3 &amp; \\ \hline 78 &amp; ✓\end{array}$</div>
<p>It does, so our answer is correct.</p>
<h2>Optional Video: Dividing Whole Numbers With a Remainder</h2>
<p>This video below by <a href="http://www.mathispower4u.com/">Mathispower4u</a> provides an example of dividing a 4 digit whole number by a 2 digit whole number with a remainder.</p>
<p><iframe allowfullscreen="allowfullscreen" frameborder="0" height="360" src="https://www.youtube.com/embed/UPUcShGCBOs?rel=0&amp;modestbranding=1&amp;cc_lang_pref=en&amp;cc_load_policy=1" width="640"></iframe></p>

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Properties of Division

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Mathematics

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Properties of Division

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