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Guides d'étude > Prealgebra

Dividing Whole Numbers

Learning Outcomes

  • Divide whole numbers and check the answer using multiplication
  • Identify and apply the division properties of one
  • Identify and apply the division properties of zero
  • Use the long division algorithm to divide multiple-digit numbers
  • Identify the divisor, dividend, and remainder in a division problem

Divide Whole Numbers

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=312\div 4=3 because 34=123\cdot 4=12. Knowing all the multiplication number facts is very important when doing division. We check our answer to division by multiplying the quotient by the divisor to determine if it equals the dividend. We know 24÷8=324\div 8=3 is correct because 38=243\cdot 8=24.

example

Divide. Then check by multiplying.
  1. 42÷642\div 6
  2. 729\frac{72}{9}
  3. 7)637\overline{)63}
Solution:
1.
42÷642\div 6
Divide 4242 by 66. 77
Check by multiplying. 767\cdot 6
4242\quad\checkmark
2.
729\frac{72}{9}
Divide 7272 by 99. 88
Check by multiplying. 898\cdot 9
7272\quad\checkmark
3.
7)637\overline{)63}
Divide 6363 by 77. 99
Check by multiplying. 979\cdot 7
6363\quad\checkmark
    What is the quotient when you divide a number by itself?

1515=1 because 115=15\frac{15}{15}=1\text{ because }1\cdot 15=15

Dividing any number (except 0)\text{(except 0)} by itself produces a quotient of 11. Also, any number divided by 11 produces a quotient of the number. These two ideas are stated in the Division Properties of One.

Division Properties of One

Any number (except 0) divided by itself is one. a÷a=1a\div a=1
Any number divided by one is the same number. a÷1=aa\div 1=a
 

example

Divide. Then check by multiplying:
  1. 11÷1111\div 11
  2. 191\frac{19}{1}
  3. 1)71\overline{)7}

Answer: Solution:

1.
11÷1111\div 11
A number divided by itself is 11. 11
Check by multiplying. 1111\cdot 11
1111\quad\checkmark
 
2.
191\frac{19}{1}
A number divided by 11 equals itself. 1919
Check by multiplying. 19119\cdot 1
1919\quad\checkmark
 
3.
1)71\overline{)7}
A number divided by 11 equals itself. 77
Check by multiplying. 717\cdot 1
77\quad\checkmark

    Suppose we have $0\text{\$0}, and want to divide it among 33 people. How much would each person get? Each person would get $0\text{\$0}. Zero divided by any number is 00. Now suppose that we want to divide $10\text{\$10} by 00. That means we would want to find a number that we multiply by 00 to get 1010. This cannot happen because 00 times any number is 00. Division by zero is said to be undefined. These two ideas make up the Division Properties of Zero.

Division Properties of Zero

Zero divided by any number is 00. 0÷a=00\div a=0
Dividing a number by zero is undefined. a÷0a\div 0 undefined
  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 00 from 10?10? Because subtracting 00 will never change the total, we will never get an answer. So we cannot divide a number by 00.

example

Divide. Check by multiplying:
  1. 0÷30\div 3
  2. 100\frac{10}{0}.

Answer: Solution

1.
0÷30\div 3
Zero divided by any number is zero. 00
Check by multiplying. 030\cdot 3
00\quad\checkmark
 
2.
10/010/0
Division by zero is undefined. undefined

 

try it

Divide. Then check by multiplying:
  When the divisor or the dividend has more than one digit, it is usually easier to use the 4)124\overline{)12} notation. This process is called long division. Let’s work through the process by dividing 7878 by 33.
Divide the first digit of the dividend, 77, by the divisor, 33.
The divisor 33 can go into 77 two times since 2×3=62\times 3=6 . Write the 22 above the 77 in the quotient. CNX_BMath_Figure_01_05_043_img-02.png
Multiply the 22 in the quotient by 22 and write the product, 66, under the77. CNX_BMath_Figure_01_05_043_img-03.png
Subtract that product from the first digit in the dividend. Subtract 767 - 6 . Write the difference, 1, under the first digit in the dividend. CNX_BMath_Figure_01_05_043_img-04.png
Bring down the next digit of the dividend. Bring down the 88. CNX_BMath_Figure_01_05_043_img-05.png
Divide 1818 by the divisor, 33. The divisor 33 goes into 1818 six times. CNX_BMath_Figure_01_05_043_img-06.png
Write 66 in the quotient above the 88.
Multiply the 66 in the quotient by the divisor and write the product, 1818, under the dividend. Subtract 1818 from 1818. CNX_BMath_Figure_01_05_043_img-07.png
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.

So 78÷3=26\text{So }78\div 3=26.

Check by multiplying the quotient times the divisor to get the dividend. Multiply 26×326\times 3 to make sure that product equals the dividend, 7878.

\begin{array}{c}\hfill \stackrel{1}{2}6\\ \hfill \underset{\text{___}}{\times 3}\\ \hfill 78 \end{array}

It does, so our answer is correct. \checkmark

Divide whole numbers

  1. Divide the first digit of the dividend by the divisor.If the divisor is larger than the first digit of the dividend, divide the first two digits of the dividend by the divisor, and so on.
  2. Write the quotient above the dividend.
  3. Multiply the quotient by the divisor and write the product under the dividend.
  4. Subtract that product from the dividend.
  5. Bring down the next digit of the dividend.
  6. Repeat from Step 1 until there are no more digits in the dividend to bring down.
  7. Check by multiplying the quotient times the divisor.
In the video below we show another example of using long division. https://youtu.be/KvVhaB5mqr8

example

Divide 2,596÷42,596\div 4. Check by multiplying:

Answer: Solution

Let's rewrite the problem to set it up for long division. CNX_BMath_Figure_01_05_044_img-01.png
Divide the first digit of the dividend, 22, by the divisor, 44. CNX_BMath_Figure_01_05_044_img-02.png
Since 44 does not go into 22, we use the first two digits of the dividend and divide 2525 by 44. The divisor 44 goes into 2525 six times.
We write the 66 in the quotient above the 55. CNX_BMath_Figure_01_05_044_img-03.png
Multiply the 66in the quotient by the divisor 44 and write the product, 2424, under the first two digits in the dividend. CNX_BMath_Figure_01_05_044_img-04.png
Subtract that product from the first two digits in the dividend. Subtract 252425 - 24 . Write the difference, 11, under the second digit in the dividend. CNX_BMath_Figure_01_05_044_img-05.png
Now bring down the 99 and repeat these steps. There are 44 fours in 1919. Write the 44 over the 99. Multiply the 44 by 44 and subtract this product from 1919. CNX_BMath_Figure_01_05_044_img-06.png
Bring down the 66 and repeat these steps. There are 99 fours in 3636. Write the 99 over the 66. Multiply the 99 by 44 and subtract this product from 3636. CNX_BMath_Figure_01_05_044_img-07.png
So 2,596÷4=6492,596\div 4=649 .
Check by multiplying. CNX_BMath_Figure_01_05_044_img-08.png
It equals the dividend, so our answer is correct.

   

example

Divide 4,506÷64,506\div 6. Check by multiplying:

Answer: Solution

Let's rewrite the problem to set it up for long division. CNX_BMath_Figure_01_05_045_img-01.png
First we try to divide 66 into 44. CNX_BMath_Figure_01_05_045_img-02.png
Since that won't work, we try 66 into 4545. There are 77 sixes in 4545. We write the 77 over the 55. CNX_BMath_Figure_01_05_045_img-03.png
Multiply the 77 by 66 and subtract this product from 4545. CNX_BMath_Figure_01_05_045_img-04.png
Now bring down the 00 and repeat these steps. There are 55 sixes in 3030. Write the 55 over the 00. Multiply the 55 by 66 and subtract this product from 3030. CNX_BMath_Figure_01_05_045_img-05.png
Now bring down the 66 and repeat these steps. There is 11 six in 66. Write the 11 over the 66. Multiply 11 by 66 and subtract this product from 66 CNX_BMath_Figure_01_05_045_img-06.png
Check by multiplying. CNX_BMath_Figure_01_05_045_img-07.png
It equals the dividend, so our answer is correct.

   

example

Divide 7,263÷97,263\div 9. Check by multiplying.

Answer: Solution

Let's rewrite the problem to set it up for long division. CNX_BMath_Figure_01_05_046_img-01.png
First we try to divide 99 into 77. CNX_BMath_Figure_01_05_046_img-02.png
Since that won't work, we try 99 into 7272. There are 88 nines in 7272. We write the 88 over the 22. CNX_BMath_Figure_01_05_046_img-03.png
Multiply the 88 by 99 and subtract this product from 7272. CNX_BMath_Figure_01_05_046_img-04.png
Now bring down the 66 and repeat these steps. There are 00 nines in 66. Write the 00 over the 66. Multiply the 00 by 99 and subtract this product from 66. CNX_BMath_Figure_01_05_046_img-05.png
Now bring down the 33 and repeat these steps. There are 77 nines in 6363. Write the 77 over the 33. Multiply the 77 by 99 and subtract this product from 6363. CNX_BMath_Figure_01_05_046_img-06.png
Check by multiplying. CNX_BMath_Figure_01_05_046_img-07.png
It equals the dividend, so our answer is correct.

  Watch this video for another example of how to use long division to divide a four digit whole number by a two digit whole number. https://youtu.be/V7Korf09iWI So far all the division problems have worked out evenly. For example, if we had 2424 cookies and wanted to make bags of 88 cookies, we would have 33 bags. But what if there were 2828 cookies and we wanted to make bags of 8?8? Start with the 2828 cookies. An image of 28 cookies placed at random. Try to put the cookies in groups of eight. An image of 28 cookies. There are 3 circles, each containing 8 cookies, leaving 3 cookies outside the circles. There are 33 groups of eight cookies, and 44 cookies left over. We call the 44 cookies that are left over the remainder and show it by writing R4 next to the 33. (The R stands for remainder.) To check this division we multiply 33 times 88 to get 2424, and then add the remainder of 44.

\begin{array}{c}\hfill 3\\ \hfill \underset{\text{___}}{\times 8}\\ \hfill 24\\ \hfill \underset{\text{___}}{+4}\\ \hfill 28\end{array}

example

Divide 1,439÷41,439\div 4. Check by multiplying.

Answer: Solution

Let's rewrite the problem to set it up for long division. CNX_BMath_Figure_01_05_047_img-01.png
First we try to divide 44 into 11. Since that won't work, we try 44 into 1414. There are 33 fours in 1414. We write the 33 over the 44. CNX_BMath_Figure_01_05_047_img-02.png
Multiply the 33 by 44 and subtract this product from 1414. CNX_BMath_Figure_01_05_047_img-03.png
Now bring down the 33 and repeat these steps. There are 55 fours in 2323. Write the 55 over the 33. Multiply the 55 by 44 and subtract this product from 2323. CNX_BMath_Figure_01_05_047_img-04.png
Now bring down the 99 and repeat these steps. There are 99 fours in 3939. Write the 99 over the 99. Multiply the 99 by 44 and subtract this product from 3939. There are no more numbers to bring down, so we are done. The remainder is 33. CNX_BMath_Figure_01_05_047_img-05.png
Check by multiplying. CNX_BMath_Figure_01_05_047_img-06.png
So 1,439÷41,439\div 4 is 359359 with a remainder of 33. Our answer is correct.

   

example

Divide and then check by multiplying: 1,461÷131,461\div 13.

Answer: Solution

Let's rewrite the problem to set it up for long division. 13)1,46113\overline{)1,461}
First we try to divide 1313 into 11. Since that won't work, we try 1313 into 1414. There is 11 thirteen in 1414. We write the 11 over the 44. CNX_BMath_Figure_01_05_048_img-02.png
Multiply the 11 by 1313 and subtract this product from 1414. CNX_BMath_Figure_01_05_048_img-03.png
Now bring down the 66 and repeat these steps. There is 11 thirteen in 1616. Write the 11 over the 66. Multiply the 11 by 1313 and subtract this product from 1616. CNX_BMath_Figure_01_05_048_img-04.png
Now bring down the 11 and repeat these steps. There are 22 thirteens in 3131. Write the 22 over the 11. Multiply the 22 by 1313 and subtract this product from 3131. There are no more numbers to bring down, so we are done. The remainder is 55. 1,462÷131,462\div 13 is 112112 with a remainder of 55. CNX_BMath_Figure_01_05_048_img-05.png
Check by multiplying. CNX_BMath_Figure_01_05_048_img-06.png
Our answer is correct.

   

example

Divide and check by multiplying: 74,521÷24174,521\div 241.

Answer: Solution

Let's rewrite the problem to set it up for long division. 241)74,521241\overline{)74,521}
First we try to divide 241241 into 77. Since that won’t work, we try 241241 into 7474. That still won’t work, so we try 241241 into745745. Since 22 divides into 77 three times, we try 33. Since 3×241=7233\times 241=723 , we write the 33 over the 55 in 745745. Note that 44 would be too large because 4×241=9644\times 241=964 , which is greater than 745745.
Multiply the 33 by 241241 and subtract this product from 745745. CNX_BMath_Figure_01_05_049_img-02.png
Now bring down the 22 and repeat these steps. 241241 does not divide into 222222. We write a 00 over the 22 as a placeholder and then continue. CNX_BMath_Figure_01_05_049_img-03.png
Now bring down the 11 and repeat these steps. Try 99. Since 9×241=2,1699\times 241=2,169 , we write the 99 over the 11. Multiply the 99 by 241241 and subtract this product from 2,2212,221. CNX_BMath_Figure_01_05_049_img-04.png
There are no more numbers to bring down, so we are finished. The remainder is 5252. So 74,521÷24174,521\div 241 is 309309 with a remainder of 5252.
Check by multiplying. CNX_BMath_Figure_01_05_049_img-05.png

  Sometimes it might not be obvious how many times the divisor goes into digits of the dividend. We will have to guess and check numbers to find the greatest number that goes into the digits without exceeding them. Watch the video below for another example of how to use long division to divide whole numbers when there is a remainder. https://youtu.be/UPUcShGCBOs  

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