Finding Remainders

Concepts Examples

We can use the following rules to find the remainder when a number is divided by another.

Rules for Finding Remainders

Rule 1 : x is divided by y When we have to divide a number by another number, then we can write x in terms of y, i.e. (y×a)+b. This means that the remainder is b.

e.g.When 66 is divided by 10, we can write 66 as 10×6+6, thus the remainder is 6.

Rule 2 : x ± y is divided by z When we have to divide the sum or difference of two number by another number, then we write both the numbers as the product of the number with which we have to divide.

e.g.When we have to divide 45+23 by 8, then we write it like

8×5+5 + 8×2+7=8(5+2)+ 12

Since 8(5+2) is divisible by 8 thus we will divide 12 by 8, and we know the remainder will be 4.

Similarily we can solve for 45-23 divided by 8.

Rule 3 : x × y is divided by z When we have to divide the product of two numbers by another number, then the remainder will be equal to the remainder of each of the number divided separately.

e.g.When 61×109 is divided by 7, the remainder of $\frac{61}{7}$617  is 5 and remainder of $\frac{109}{7}$1097  is 4, thus we can say the remainder will be $\frac{\left(5\cdot4\right)}{7}=\frac{20}{7}=6$(5·4)7 =207 =6

Rule 4 : x^y is divided by z When we have to divide a number to some power with another number, then keep dividing the number with the divisor till the remainder is not less than the divisor.

e.g. When 7⁴⁷ is divided by 5, $\frac{7}{5}$75  will give the remainder as 2, thus 7⁴⁷ divided by 5 will give remainder as

$\frac{2\cdot2\cdot2...2}{5}$2·2·2...25  i.e 47 times 2.

Now, 2⁴⁷ = 2³×2⁴¹¹=$\frac{8\cdot16^{11}}{5}$8·16¹¹5 

$\frac{8}{5}$85  gives the remainder as 3 and $\frac{16}{5}$165  gives the remainder as 1.

∴ $\frac{7^{47}}{5}$7⁴⁷5  gives the remainder as 3.

Rule of Negative Remainder When a number x divided by y, then we can say that the remainder is -a if y×k=x+a, where k can be any natural number and a is smaller than y. This in turn means that the remainder is a times smaller than y.

e.g When 8 is divided by 5, we can that remainder is -2. It means 2 less than the divisor i.e 3

e.g When 120 is divided by 11, we can that remainder is -1. It means 1 less than the divisor i.e 10

Question 1 What will be remainder when 3^97! is divided by 80

2	1
3	4

B

3^97! can be written as 81^k

$\frac{81^k}{80}$81k80 

Thus the remainder is 1

Question 2 What will be remainder when 1234 × 5678 is divided by 11

1	2
4	6

C

Using Rule 3, Remainder of

$\frac{1234\times5678}{11}$1234×567811 

= Remainder of

$\frac{2\times2}{11}$2×211 

=4

Question 3 What will be remainder when 75⁷⁵⁷⁵ is divided by 37

4	3
2	1

D

75 when divided by 37 leaves the remainder as 1 ,

Thus the remainder when 75⁷⁵⁷⁵ is divided by 37 is 1

Question 4 What will be remainder when 1!+2!+3!+4!..100! is divided by 7

4	5
6	7

B

All the numbers from 7! to 100! are divisible by 7, since they contains atleast 1 factor as 7

1!+2!+3!+4!+5!+6! when divided by 7 will give remainder

$\frac{1+2+6+24+120+720}{7}$1+2+6+24+120+7207 

=

$\frac{1+2+-1+3+1+6}{7}=\frac{12}{7}=5$1+2+-1+3+1+67 =127 =5

Thus the remainder is 5

Question 5 What will be remainder when 7⁴⁸⁶ is divided by 4

0	1
2	3

D

$\frac{7}{4}$74 

will give the remainder as -1,

∴

$\frac{7^{486}}{4}$74864 

will give remainder as

$\frac{-1^{486}}{4}$-14864 

= -1

Thus the remainder will be 3

Question 6 What will be remainder when 1234+5678 is divided by 5

1	2
3	4

B

Using Rule 2,

$\frac{1234}{5}$12345 

gives remainder 4 and

$\frac{5678}{5}$56785 

gives remainder 3

thus the remainder will be

$\frac{4+3}{5}=2$4+35 =2

Question 7 What will be remainder when 1719 × 1721 × 1723 × 1725 × 1727 is divided by 18

3	6
8	9

D

Using Rule 3,

The remainder of

$\frac{1719\times1721\times1723\times1725\times1727}{18}$1719×1721×1723×1725×172718 

= Remainder of

$\frac{9\times11\times13\times15\times17}{18}$9×11×13×15×1718 

= Remainder of

$\frac{99\times13\times15\times17}{18}$99×13×15×1718 

= Remainder of

$\frac{9\times13\times255}{18}$9×13×25518 

= Remainder of

$\frac{117\times255}{18}=\frac{9\times3}{18}=\frac{27}{18}$117×25518 =9×318 =2718 

= 9