Division Algorithm
Key Concepts
Solved Examples: 22
Practice Questions: 3
Solved Examples
Question 1: When a number is divided by 24, the quotient is 18 and the remainder is 11. Find the dividend.
Using the Division Algorithm,
Dividend = Divisor × Quotient + Remainder
Substituting the given values,
Dividend = 24 × 18 + 11
= 432 + 11
= 443
Dividend = Divisor × Quotient + Remainder
Substituting the given values,
Dividend = 24 × 18 + 11
= 432 + 11
= 443
Question 2: A number is divided by a certain divisor. If the quotient is 28, the remainder is 9, and the dividend is 821, find the divisor.
Using the Division Algorithm,
Dividend = Divisor × Quotient + Remainder
Substituting the given values,
821 = Divisor × 28 + 9
Subtract 9 from both sides.
812 = Divisor × 28
Divide both sides by 28.
Divisor = 812 ÷ 28
= 29
Dividend = Divisor × Quotient + Remainder
Substituting the given values,
821 = Divisor × 28 + 9
Subtract 9 from both sides.
812 = Divisor × 28
Divide both sides by 28.
Divisor = 812 ÷ 28
= 29
Question 3: The dividend is 2579, the divisor is 51, and the remainder is 29. Find the quotient.
Using the Division Algorithm,
2579 = 51 × Quotient + 29
Subtract 29 from both sides.
2550 = 51 × Quotient
Quotient = 2550 ÷ 51
= 50
2579 = 51 × Quotient + 29
Subtract 29 from both sides.
2550 = 51 × Quotient
Quotient = 2550 ÷ 51
= 50
Question 4: A number is divided by 37. If the dividend is 1429 and the quotient is 38, find the remainder.
Using the Division Algorithm,
Dividend = Divisor × Quotient + Remainder
Substituting the given values,
1429 = 37 × 38 + Remainder
= 1406 + Remainder
Remainder = 1429 − 1406
= 23
Since 23 less than 37, the remainder is valid.
Dividend = Divisor × Quotient + Remainder
Substituting the given values,
1429 = 37 × 38 + Remainder
= 1406 + Remainder
Remainder = 1429 − 1406
= 23
Since 23 less than 37, the remainder is valid.
Question 5: A student claims that when 1847 is divided by 36, the quotient is 51 and the remainder is 11. Verify whether the division is correct.
Using the Division Algorithm,
Dividend = Divisor × Quotient + Remainder
= 36 × 51 + 11
= 1836 + 11
= 1847
The calculated dividend matches the given dividend.
Dividend = Divisor × Quotient + Remainder
= 36 × 51 + 11
= 1836 + 11
= 1847
The calculated dividend matches the given dividend.
Question 6: A factory packs 2,875 chocolates into boxes, each containing 36 chocolates. How many completely filled boxes can be made, and how many chocolates will be left over?
Dividend = 2875
Divisor = 36
Quotient = 2875 ÷ 36 = 79
36 × 79 = 2844
Remainder = 2875 − 2844 = 31
Therefore, number of completely filled boxes = 79
Chocolates left over = 31
Divisor = 36
Quotient = 2875 ÷ 36 = 79
36 × 79 = 2844
Remainder = 2875 − 2844 = 31
Therefore, number of completely filled boxes = 79
Chocolates left over = 31
Question 7: In a division problem, the divisor is 5 times the remainder. If the quotient is 18 and the remainder is 7, find the dividend.
The divisor is 5 times the remainder.
Divisor = 5 × 7 = 35
Using the Division Algorithm,
Dividend = Divisor × Quotient + Remainder
= 35 × 18 + 7
= 630 + 7
= 637
Divisor = 5 × 7 = 35
Using the Division Algorithm,
Dividend = Divisor × Quotient + Remainder
= 35 × 18 + 7
= 630 + 7
= 637
Question 8: In a division problem, the divisor is 4 times the quotient and 3 times the remainder. If the remainder is 4, find the dividend.
The remainder is 4.
Since the divisor is 3 times the remainder,
Divisor = 3 × 4 = 12
Also, the divisor is 4 times the quotient.
Therefore,
Quotient = 12 ÷ 4 = 3
Now, using the Division Algorithm,
Dividend = Divisor × Quotient + Remainder
= 12 × 3 + 4
= 36 + 4
= 40
Since the divisor is 3 times the remainder,
Divisor = 3 × 4 = 12
Also, the divisor is 4 times the quotient.
Therefore,
Quotient = 12 ÷ 4 = 3
Now, using the Division Algorithm,
Dividend = Divisor × Quotient + Remainder
= 12 × 3 + 4
= 36 + 4
= 40
Question 9: What is the smallest number that should be added to 1000 so that the resulting number is exactly divisible by 45?
First, divide 1000 by 45.
1000 = 45 × 22 + 10
The remainder is 10.
To make the number exactly divisible by 45, we need to add the difference between the divisor and the remainder.
Required number = 45 − 10 = 35
1000 = 45 × 22 + 10
The remainder is 10.
To make the number exactly divisible by 45, we need to add the difference between the divisor and the remainder.
Required number = 45 − 10 = 35
Question 10: A certain number leaves a remainder of 39 when divided by 357. What remainder will it leave when divided by 17?
Since the remainder is 39 when the number is divided by 357,
N = 357k + 39
Now,
357 = 17 × 21
and
39 = 17 × 2 + 5
Therefore,
N = 17 × 21k + (17 × 2 + 5)
= 17(21k + 2) + 5
Hence, when N is divided by 17, the remainder is 5.
N = 357k + 39
Now,
357 = 17 × 21
and
39 = 17 × 2 + 5
Therefore,
N = 17 × 21k + (17 × 2 + 5)
= 17(21k + 2) + 5
Hence, when N is divided by 17, the remainder is 5.
Question 11: A certain number leaves a remainder of 39 when divided by 357. What remainder will it leave when divided by 17?
Let the required number be N.
Since the remainder is 63 when the number is divided by 899,
N = 899k + 63
Now,
899 = 29 × 31
and
63 = 29 × 2 + 5
Therefore,
N = 29 × 31k + (29 × 2 + 5)
= 29(31k + 2) + 5
Hence, when N is divided by 29, the remainder is 5.
Since the remainder is 63 when the number is divided by 899,
N = 899k + 63
Now,
899 = 29 × 31
and
63 = 29 × 2 + 5
Therefore,
N = 29 × 31k + (29 × 2 + 5)
= 29(31k + 2) + 5
Hence, when N is divided by 29, the remainder is 5.
Question 12: A number when divided by 12 leaves a certain quotient and a remainder of 5. If the same number is divided by 15, the new quotient is 4 less than the original quotient, and the remainder is 8. Find the number.
Let the original quotient be q.
When the number is divided by 12, the remainder is 5. Therefore, Number = 12q + 5
When the same number is divided by 15, the new quotient is 4 less, i.e. q − 4, and the remainder is 8.
Therefore Number = 15(q − 4) + 8
Since both expressions represent the same number:
12q + 5 = 15(q − 4) + 8
or, 12q + 5 = 15q − 60 + 8
or, 12q + 5 = 15q − 52
or, 57 = 3q
or, q = 19
Therefore, the number is 12 × 19 + 5 = 228 + 5 = 233
When the number is divided by 12, the remainder is 5. Therefore, Number = 12q + 5
When the same number is divided by 15, the new quotient is 4 less, i.e. q − 4, and the remainder is 8.
Therefore Number = 15(q − 4) + 8
Since both expressions represent the same number:
12q + 5 = 15(q − 4) + 8
or, 12q + 5 = 15q − 60 + 8
or, 12q + 5 = 15q − 52
or, 57 = 3q
or, q = 19
Therefore, the number is 12 × 19 + 5 = 228 + 5 = 233
Question 13: In a division problem, the quotient is 12 times the remainder, and the divisor is 3 times the quotient. If the sum of the quotient and the remainder is 65, find the dividend.
Let the remainder be r.
The quotient is 12 times the remainder q = 12r
The divisor is 3 times the quotient d = 3q = 36r
We are given q + r = 65
Substituting q = 12r
12r + r = 65
or, 13r = 65
or, r = 5
Therefore q = 12 × 5 = 60
and
d = 3 × 60 = 180
Using the Division Algorithm:
Dividend = Divisor × Quotient + Remainder
= 180 × 60 + 5
= 10,805
The quotient is 12 times the remainder q = 12r
The divisor is 3 times the quotient d = 3q = 36r
We are given q + r = 65
Substituting q = 12r
12r + r = 65
or, 13r = 65
or, r = 5
Therefore q = 12 × 5 = 60
and
d = 3 × 60 = 180
Using the Division Algorithm:
Dividend = Divisor × Quotient + Remainder
= 180 × 60 + 5
= 10,805
Question 14: A number is successively divided by 4 and 5. This means the original number is divided by 4 to get a first quotient, and then that first quotient is divided by 5. If the remainders are 1 and 4 respectively, find the remainder when the original number is divided by 20.
Let the original number be N.
When N is divided by 4, the remainder is 1. Let the first quotient be q.
Therefore N = 4q + 1
Now q is divided by 5, leaving a remainder of 4. Let the second quotient be k.
Therefore q = 5k + 4
Substitute this into the first equation:
N = 4(5k + 4) + 1
N = 20k + 16 + 1
N = 20k + 17
Therefore, when N is divided by 20, the remainder is 17
When N is divided by 4, the remainder is 1. Let the first quotient be q.
Therefore N = 4q + 1
Now q is divided by 5, leaving a remainder of 4. Let the second quotient be k.
Therefore q = 5k + 4
Substitute this into the first equation:
N = 4(5k + 4) + 1
N = 20k + 16 + 1
N = 20k + 17
Therefore, when N is divided by 20, the remainder is 17
Question 15: Find the largest 5-digit number that is exactly divisible by 124.
The largest 5-digit number is 99,999.
We need the largest multiple of 124 that does not exceed 99,999.
Divide 99,999 ÷ 124 = 806 remainder 55
Therefore, the largest exact multiple is:
124 × 806 = 99,944
We need the largest multiple of 124 that does not exceed 99,999.
Divide 99,999 ÷ 124 = 806 remainder 55
Therefore, the largest exact multiple is:
124 × 806 = 99,944
Question 16: Two numbers when divided by the same divisor leave remainders of 432 and 347 respectively. When the sum of these two numbers is divided by the same divisor, the remainder is 211. Find the value of the divisor.
Let the divisor be d.
The two numbers leave remainders 432 and 347 when divided by d.
Therefore, their sum leaves a remainder of
432 + 347 = 779
But the actual remainder is 211. Hence, the excess 779 − 211 = 568 must be exactly divisible by the divisor.
Therefore, d is a divisor of 568.
Now factorise 568 = 8 × 71
The possible divisors greater than 432 (because a remainder must always be less than the divisor) are 568
∴ Divisor = 568
The two numbers leave remainders 432 and 347 when divided by d.
Therefore, their sum leaves a remainder of
432 + 347 = 779
But the actual remainder is 211. Hence, the excess 779 − 211 = 568 must be exactly divisible by the divisor.
Therefore, d is a divisor of 568.
Now factorise 568 = 8 × 71
The possible divisors greater than 432 (because a remainder must always be less than the divisor) are 568
∴ Divisor = 568
Question 17: A number, when divided by 7, leaves a remainder of 3 and gives a quotient of 14. If the number is multiplied by 5 and the resulting number is then divided by 7, find the new quotient and the new remainder.
Let the original number be N.
Using the Division Algorithm:
N = 7 × 14 + 3
N = 98 + 3 = 101
Now multiply the number by 5:
5N = 5 × 101 = 505
Divide 505 by 7:
505 = 7 × 72 + 1
Therefore, the new quotient is 72 and the new remainder is 1.
∴ New quotient = 72
∴ New remainder = 1
Using the Division Algorithm:
N = 7 × 14 + 3
N = 98 + 3 = 101
Now multiply the number by 5:
5N = 5 × 101 = 505
Divide 505 by 7:
505 = 7 × 72 + 1
Therefore, the new quotient is 72 and the new remainder is 1.
∴ New quotient = 72
∴ New remainder = 1
Question 18: A number is successively divided by 3, 4, and 5, leaving remainders of 2, 1, and 3, respectively. If the quotient obtained after the final division is 2, find the original number.
Work backwards using the Division Algorithm.
On the final division by 5:
Second Quotient = 5 × 2 + 3 = 13
Now 13 is the quotient obtained from the second division by 4:
First Quotient = 4 × 13 + 1 = 53
Now 53 is the quotient obtained from the first division by 3:
Original Number = 3 × 53 + 2
= 159 + 2
= 161
Verification
161 ÷ 3 = 53 remainder 2
53 ÷ 4 = 13 remainder 1
13 ÷ 5 = 2 remainder 3
On the final division by 5:
Second Quotient = 5 × 2 + 3 = 13
Now 13 is the quotient obtained from the second division by 4:
First Quotient = 4 × 13 + 1 = 53
Now 53 is the quotient obtained from the first division by 3:
Original Number = 3 × 53 + 2
= 159 + 2
= 161
Verification
161 ÷ 3 = 53 remainder 2
53 ÷ 4 = 13 remainder 1
13 ÷ 5 = 2 remainder 3
Question 19: A natural number N, when divided by a certain divisor, leaves a remainder of 24. When 3N is divided by the same divisor, the remainder is 11. Find the divisor.
Let the divisor be d.
Since N leaves a remainder of 24 when divided by d, by the Division Algorithm:
N = dq + 24
where q is the quotient.
Multiplying by 3:
3N = 3dq + 72
Therefore, when 3N is divided by d, the remainder is the remainder obtained when 72 is divided by d.
We are told that this remainder is 11.
Hence 72 − 11 = 61 must be exactly divisible by d.
Therefore, d must be a divisor of 61.
Since the original remainder is 24, the divisor must be greater than 24.
As 61 is prime, its only positive divisors are 1 and 61. The only one greater than 24 is 61.
∴ Divisor = 61
Quick check
If N = 61q + 24, then:
3N = 183q + 72
= 61(3q + 1) + 11
So the remainder is indeed 11.
Since N leaves a remainder of 24 when divided by d, by the Division Algorithm:
N = dq + 24
where q is the quotient.
Multiplying by 3:
3N = 3dq + 72
Therefore, when 3N is divided by d, the remainder is the remainder obtained when 72 is divided by d.
We are told that this remainder is 11.
Hence 72 − 11 = 61 must be exactly divisible by d.
Therefore, d must be a divisor of 61.
Since the original remainder is 24, the divisor must be greater than 24.
As 61 is prime, its only positive divisors are 1 and 61. The only one greater than 24 is 61.
∴ Divisor = 61
Quick check
If N = 61q + 24, then:
3N = 183q + 72
= 61(3q + 1) + 11
So the remainder is indeed 11.
Question 20: Find the smallest 5-digit number that must be added to 45,678 so that the resulting sum is exactly divisible by 312.
First, divide 45,678 by 312:
45,678 = 312 × 146 + 126
So, the remainder is 126.
To make the resulting number exactly divisible by 312, the number added must leave a remainder of:
312 − 126 = 186
Therefore, the required number is of the form:
312k + 186
We need the smallest 5-digit number.
Since the number must be at least 10,000:
312k + 186 >= 10,000
312k >= 9,814
k >= 31.455…
Thus, the smallest possible integer value of k is 32.
Therefore:
312 × 32 + 186 = 9,984 + 186 = 10,170
Check:
45,678 + 10,170 = 55,848 and
55,848 ÷ 312 = 179
So the sum is exactly divisible by 312.
So the smallest 5 digit number is 10,170
45,678 = 312 × 146 + 126
So, the remainder is 126.
To make the resulting number exactly divisible by 312, the number added must leave a remainder of:
312 − 126 = 186
Therefore, the required number is of the form:
312k + 186
We need the smallest 5-digit number.
Since the number must be at least 10,000:
312k + 186 >= 10,000
312k >= 9,814
k >= 31.455…
Thus, the smallest possible integer value of k is 32.
Therefore:
312 × 32 + 186 = 9,984 + 186 = 10,170
Check:
45,678 + 10,170 = 55,848 and
55,848 ÷ 312 = 179
So the sum is exactly divisible by 312.
So the smallest 5 digit number is 10,170
Question 21: How many three-digit natural numbers, when divided by 14, give a quotient that is exactly twice the remainder?
Let the remainder be r.
Since the quotient is twice the remainder q = 2r
Using the Division Algorithm N = 14q + r
Substituting q = 2r:
N = 14(2r) + r
N = 29r
Since the remainder must be less than the divisor 0 <= r < 14
Also, N is a three-digit number 100 <= 29r <= 999
Therefore 100/29 <= r <= 999/29
3.45… <= r <= 34.44…
Combining this with 0 <= r < 14, the possible positive integer zalues of r are
r = 4, 5, 6, …, 13 That’s 10 values.
Each value gives a different three-digit number because N = 29r.
For example:
r = 4 → N = 116
r = 5 → N = 145
r = 13 → N = 377
∴ Number of such three-digit natural numbers = 10
Since the quotient is twice the remainder q = 2r
Using the Division Algorithm N = 14q + r
Substituting q = 2r:
N = 14(2r) + r
N = 29r
Since the remainder must be less than the divisor 0 <= r < 14
Also, N is a three-digit number 100 <= 29r <= 999
Therefore 100/29 <= r <= 999/29
3.45… <= r <= 34.44…
Combining this with 0 <= r < 14, the possible positive integer zalues of r are
r = 4, 5, 6, …, 13 That’s 10 values.
Each value gives a different three-digit number because N = 29r.
For example:
r = 4 → N = 116
r = 5 → N = 145
r = 13 → N = 377
∴ Number of such three-digit natural numbers = 10
Question 22: How many three-digit natural numbers, when divided by 17, give a quotient that is 3 more than twice the remainder?
Let the remainder be r.
According to the Division Algorithm N = 17q + r
We are told that the quotient is 3 more than twice the remainder q = 2r + 3
Substituting:
N = 17(2r + 3) + r
N = 34r + 51 + r
N = 35r + 51
Since the remainder must be less than the divisor 0 <= r < 17
Also, N is a three-digit number 100 <= 35r + 51 <= 999
From the lower bound 35r + 51 >= 100
35r >= 49
r >= 1.4
Thus, r >= 2
The upper bound gives
35r + 51 <= 999
35r <= 948
r <= 27.08
But we already know that r < 17.
Therefore, the possible values are
r = 2, 3, 4, …, 16
Number of possible values
16 − 2 + 1 = 15
According to the Division Algorithm N = 17q + r
We are told that the quotient is 3 more than twice the remainder q = 2r + 3
Substituting:
N = 17(2r + 3) + r
N = 34r + 51 + r
N = 35r + 51
Since the remainder must be less than the divisor 0 <= r < 17
Also, N is a three-digit number 100 <= 35r + 51 <= 999
From the lower bound 35r + 51 >= 100
35r >= 49
r >= 1.4
Thus, r >= 2
The upper bound gives
35r + 51 <= 999
35r <= 948
r <= 27.08
But we already know that r < 17.
Therefore, the possible values are
r = 2, 3, 4, …, 16
Number of possible values
16 − 2 + 1 = 15
What is Division Algorithm?
The Division Algorithm is a mathematical rule that expresses the relationship between the dividend, divisor, quotient, and remainder when one positive integer is divided by another. It states that for any integers a and b, where b > 0, there exist unique integers q and r such that:
a = bq + r
where:
a = Dividend
b = Divisor
q = Quotient
r = Remainder
The remainder must satisfy the condition 0 ≤ r < b In other words, the remainder can be 0 or any positive integer smaller than the divisor, but it can never be equal to or greater than the divisor. For example, when 47 is divided by 6: 47 = 6 × 7 + 5 Here, a = 47, b = 6, q = 7, and r = 5. Since 0 ≤ 5 < 6, the Division Algorithm is satisfied. The Division Algorithm can be used to find unknown quantities in a division, verify a division, and solve problems involving remainders and divisibility. The Euclidean Algorithm is a systematic method of finding the HCF (or GCD) of two positive integers by applying the Division Algorithm repeatedly. At each step, the divisor becomes the next dividend, and the remainder becomes the next divisor. The process continues until the remainder becomes 0. The last non-zero remainder is the HCF/GCD of the two numbers. Suppose we want to find the HCF of 252 and 198. Applying the Division Algorithm repeatedly: 252 = 198 × 1 + 54 198 = 54 × 3 + 36 54 = 36 × 1 + 18 36 = 18 × 2 + 0 Since the last non-zero remainder is 18, ∴ HCF(252, 198) = 18 The process can be summarised as follows:
a = bq + r
where:
a = Dividend
b = Divisor
q = Quotient
r = Remainder
The remainder must satisfy the condition 0 ≤ r < b In other words, the remainder can be 0 or any positive integer smaller than the divisor, but it can never be equal to or greater than the divisor. For example, when 47 is divided by 6: 47 = 6 × 7 + 5 Here, a = 47, b = 6, q = 7, and r = 5. Since 0 ≤ 5 < 6, the Division Algorithm is satisfied. The Division Algorithm can be used to find unknown quantities in a division, verify a division, and solve problems involving remainders and divisibility. The Euclidean Algorithm is a systematic method of finding the HCF (or GCD) of two positive integers by applying the Division Algorithm repeatedly. At each step, the divisor becomes the next dividend, and the remainder becomes the next divisor. The process continues until the remainder becomes 0. The last non-zero remainder is the HCF/GCD of the two numbers. Suppose we want to find the HCF of 252 and 198. Applying the Division Algorithm repeatedly: 252 = 198 × 1 + 54 198 = 54 × 3 + 36 54 = 36 × 1 + 18 36 = 18 × 2 + 0 Since the last non-zero remainder is 18, ∴ HCF(252, 198) = 18 The process can be summarised as follows:
| Step | Division | Remainder |
|---|---|---|
| 1 | 252 = 198 × 1 + 54 | 54 |
| 2 | 198 = 54 × 3 + 36 | 36 |
| 3 | 54 = 36 × 1 + 18 | 18 |
| 4 | 36 = 18 × 2 + 0 | 0 |
| Last non-zero remainder = HCF = 18 | ||
The Euclidean Algorithm does not introduce a new division rule. It simply repeats the Division Algorithm until the remainder becomes 0.
Summary of Division Algorithm
| Concept | Rule / Formula | Key Point |
|---|---|---|
| Division Algorithm | Dividend = Divisor × Quotient + Remainder | Relates the four terms in every division. |
| Dividend | Number being divided | Usually the largest number in the division. |
| Divisor | Number by which the dividend is divided | Must be greater than the remainder. |
| Quotient | Result of the division | Can be found using the division formula. |
| Remainder | 0 ≤ Remainder < Divisor | Always non-negative and smaller than the divisor. |
| Exact Division | Remainder = 0 | The dividend is exactly divisible by the divisor. |
| Remember: The remainder is always less than the divisor. | ||
Common Mistakes
- Using the wrong formula: Always use the Division Algorithm correctly: Dividend = Divisor × Quotient + Remainder.
- Forgetting that the remainder must be smaller than the divisor: The remainder is always greater than or equal to 0 and less than the divisor.
- Interchanging the dividend and divisor: Remember that the dividend is the number being divided, while the divisor is the number by which it is divided.
- Ignoring the remainder while finding the dividend: Do not forget to add the remainder after multiplying the divisor and quotient.
- Making errors while rearranging the formula: When finding the divisor or quotient, rearrange the equation carefully before substituting the values.
- Not checking the final answer: Verify your answer by substituting the values back into the Division Algorithm.
- Using a negative remainder: In elementary division problems, the remainder is never negative.
- Assuming every division has a remainder: If the dividend is exactly divisible by the divisor, the remainder is 0.
- Misreading word problems: Identify the dividend, divisor, quotient and remainder correctly before applying the formula.
Practice Questions
Question 1: In a division problem, the quotient is 7 less than the divisor and the remainder is 9. If the dividend is 525, find the divisor.
Question 2: A number is divided by 56. The quotient obtained is 24 and the remainder is r. If the dividend is 1370, determine whether the division is possible. If possible, find the remainder; otherwise, explain why it is not possible.
Question 3: A positive integer, when divided by 39, gives a quotient equal to the remainder. Find the greatest possible dividend.
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