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Not all divisions end in zero. We will examine such divisions in a subsequent subsection.

Practice set a

Perform the following divisions.

126 ÷ 7 size 12{"126" div 7} {}

18

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324 ÷ 4 size 12{"324" div 4} {}

81

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2, 559 ÷ 3 size 12{2,"559" div 3} {}

853

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5, 645 ÷ 5 size 12{5,"645" div 5} {}

1,129

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757 , 125 ÷ 9 size 12{"757","125" div 9} {}

84,125

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Division with a multiple digit divisor

The process of division also works when the divisor consists of two or more digits. We now make educated guesses using the first digit of the divisor and one or two digits of the dividend.

Sample set b

Find 2, 232 ÷ 36 size 12{2,"232" div "36"} {} .

36 2232

Use the first digit of the divisor and the first two digits of the dividend to make the educated guess.

3 goes into 22 at most 7 times.

Try 7: 7 × 36 = 252 size 12{7 times "36"="252"} {} which is greater than 223. Reduce the estimate.

Try 6: 6 × 36 = 216 size 12{6 times "36"="216"} {} which is less than 223.

The first step of a long division problem. 2232 divided by 36. 36 goes into 223 approximately 6 times, with a remainder of 7. The ones digit of 2232 is then brought down to adjoin the 7. Multiply: 6 × 36 = 216. Write 216 below 223. Subtract: 223 - 216 = 7. Bring down the 2.

Divide 3 into 7 to estimate the number of times 36 goes into 72. The 3 goes into 7 at most 2 times.

Try 2: 2 × 36 = 72 size 12{2 times "36"="72"} {} .

Long division. 2232 divided by 36. 36 goes into 223 approximately 6 times, with a remainder of 7. The ones digit of 2232 is then brought down to adjoin the 7. 36 goes into 72 exactly twice, leaving a remainder of 0.

Check : Is 2232 equal to 36 time 62? Yes.

Thus, 2, 232 ÷ 36 = 62 size 12{2,"232" div "36"="62"} {} .

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Find 2, 417 , 228 ÷ 802 size 12{2,"417","228" div "802"} {} .

802 2417228

First, the educated guess: 24 ÷ 8 = 3 size 12{"24" div 8=3} {} . Then 3 × 802 = 2406 size 12{3 times "802"="2406"} {} , which is less than 2417. Use 3 as the guess. Since 3 × 802 = 2406 size 12{3 times "802"="2406"} {} , and 2406 has four digits, place the 3 above the fourth digit of the dividend.

The first step of a long division problem. 2417228 divided by 802. 802 goes into 2417 approximately 3 times, with a remainder of 11. The hundreds digit of 2417228 is then brought down to adjoin the 11.
Subtract: 2417 - 2406 = 11.
Bring down the 2.

The divisor 802 goes into 112 at most 0 times. Use 0.

The second step of a long division problem. 802 goes into 112 0 times, so a zero is placed above, and the next digit is brought down.
Multiply: 0 × 802 = 0. Subtract: 112 - 0 = 112. Bring down the 2.

The 8 goes into 11 at most 1 time, and 1 × 802 = 802 size 12{1 times "802"="802"} {} , which is less than 1122. Try 1.

The third step of a long division problem. 802 goes into 1122 once, so a 1 is placed above and the ones digit is brought down.
Subtract 1122 - 802 = 320
Bring down the 8.

8 goes into 32 at most 4 times.

4 × 802 = 3208 size 12{4 times "802"="3208"} {} .

Use 4.

The third step of a long division problem. 2417228 divided by 802. 802 goes into 2417 approximately 3 times, with a remainder of 11. The hundreds digit of 2417228 is then brought down to adjoin the 11. 802 goes into 112 0 times, so a zero is placed above, and the next digit is brought down. 802 goes into 1122 once, so a 1 is placed above and the ones digit is brought down. 802 goes into 3208 4 times, leaving a remainder of 0.

Check: Is 2417228 equal to 3014 times 802? Yes.

Thus, 2, 417 , 228 ÷ 802 = 3, 014 size 12{2,"417","228" div "802"=3,"014"} {} .

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Practice set b

Perform the following divisions.

1, 376 ÷ 32 size 12{1,"376" div "32"} {}

43

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6, 160 ÷ 55 size 12{6,"160" div "55"} {}

112

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18 , 605 ÷ 61 size 12{"18","605" div "61"} {}

305

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144 , 768 ÷ 48 size 12{"144","764" div "48"} {}

3,016

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Division with a remainder

We might wonder how many times 4 is contained in 10. Repeated subtraction yields

10 -   4 ̲ 6 - 4 ̲ 2

Since the remainder is less than 4, we stop the subtraction. Thus, 4 goes into 10 two times with 2 remaining. We can write this as a division as follows.

2 4 10 -   8 ̲ 2
Divide: 4 goes into 10 at most 2 times. Multiply: 2 × 4 = 8. Write 8 below 0. Subtract: 10 - 8 = 2.

Since 4 does not divide into 2 (the remainder is less than the divisor) and there are no digits to bring down to continue the process, we are done. We write

2 R 2 4    10 - 8 ̲ 2 or 10 ÷ 4 = 2 R 2 2 with remainder 2

Sample set c

Find 85 ÷ 3 size 12{"85" div 3} {} .

Long division. 85 divided by 3. 3 goes into 8 twice, with a remainder of 2. The ones digit is then brought down. 3 goes into 25 8 times, with a remainder of 1.
Divide: 3 goes into 8 at most 2 times. Multiply: 2 × 3 = 6. Write 6 below 8. Subtract: 8 - 6 = 2. Bring down the 5. Divide: 3 goes into 25 at most 8 times. Multiply: 3 × 8 = 24. Write 24 below 25. Subtract: 25 - 24 = 1.

There are no more digits to bring down to continue the process. We are done. One is the remainder.

Check: Multiply 28 and 3, then add 1.

28 ×   3 ̲ 84 +   1 ̲ 85

Thus, 85 ÷ 3 = 28 R1 size 12{"85" div 3="28"`R1} {} .

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Find 726 ÷ 23 size 12{"726" div "23"} {} .

Long division. 726 divided by 23. 23 goes into 72 three times, with a remainder of 3. The ones digit is then brought down. 23 goes into 36 once, with a remainder of 13.

Check: Multiply 31 by 23, then add 13.

31 times 23 equals 713. 713 plus 13 equals 726.

Thus, 726 ÷ 23 = 31 R 13 size 12{"726" div "23"="31"````R"13"} {} .

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Practice set c

Perform the following divisions.

75 ÷ 4 size 12{"75" div 4} {}

18 R3

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346 ÷ 8 size 12{"346" div 8} {}

43 R2

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489 ÷ 21 size 12{"489" div "21"} {}

23 R6

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5, 016 ÷ 82 size 12{5,"016" div "82"} {}

61 R14

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41 , 196 ÷ 67 size 12{"41","196" div "67"} {}

614 R58

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Calculators

The calculator can be useful for finding quotients with single and multiple digit divisors. If, however, the division should result in a remainder, the calculator is unable to provide us with the particular value of the remainder. Also, some calculators (most nonscientific) are unable to perform divisions in which one of the numbers has more than eight digits.

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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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Graphene has a hexagonal structure
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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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