This is the final article covering some more handy methods to find remainders, out of which the most comfortable method can be chosen to save time dealing with remainder questions.

**Method 1: **

**Example**: Suppose we have two numbers “a” and “b”. If “a” is in form of 7n +4 and “b” is in form of 7m+1.

**Addition**

Find remainder when a+b is divided by 7.

Given a = 7n +4

and b = 7m + 1

So when we add them we will have a +b = 7n + 7m + 4 + 1 = 7 (n + m) + 5

That means we have a multiple of 7 and a remainder 5, because five is smaller than the 7. So it cannot be divided further and hence this 5 is our remainder.

** ****Difference**

What will be the remainder when a – b is divided by 7?

Given a = 7n +4

and b = 7m + 1

So when we subtract them we will have a – b= (7n +4) – (7m + 1)

= 7(n – m ) + 4 – 1

= 7(n – m ) + 3.

That means we will have a multiple of 7 with addition of 3 and because three

is smaller than the 7 so it cannot be divided further and hence this 3 is our remainder .

** **

**Multiplication**

What will be the remainder when a x b is divided by 7?

Given: a = 7n +4

and b = 7m + 1

So when we multiply them we will have a x b = (7n +4) x (7m + 1)

= 7n(7m + 1) + 4(7m + 1)

= 49nm + 7n + 28nm + 4. Here we can see that each and every number is a multiple of 7 except 4. Hence this 4 is our remainder.

**Some Important Properties of Prime Numbers**

- Any single digit number written (
*P*– 1) times is divisible by*P*, where*P*is a prime number > 5.

**Example: **Find the remainder of

**Solution:**

11 is a prime number. Any digit repeated (11-1) times or multiple of 10 times must be divisible by 11.

Now, in the given question, digit 7 is repeated 99 times hence it must be divisible by 11 .so remainder = 0

**Example: **Find the remainder of

**Solution:**

19 is a prime number and any digit repeated (19-1) times or multiple of 18 times must be divisible by 19.

**Method 2: Cancelation **

Cancellation is applied to make the calculations easier. First we cancel out the common factors and then multiply those factors back after finding remainders.

**Example: **Find the remainder of

**Solution:**

Method 1:

Since common factor is 9 = 3^{2}

Try out some more questions based on this concept so as to get a good hold of this topic.

* *

**EXERCISE: **

**Question 1.** What is the remainder when 111 + 68 + 48 is divided by 11?

a) 5

b) 7

c) 9

d) None

### Answer and Explanation

**Solution: option b**
111 leaves remainder 1 when divided by 11

68 leaves remainder 2 when divided by 11

48 leaves remainder 4 when divided by 11

Thus111 + 68 + 48 leaves remainder 1+2+4 =7 when divided by 11

**Question 2.** Find the remainder of

a)5

b)0

c)9

d)None

### Answer and Explanation

7 is a prime number. Any digit repeated (7-1)=6 times or multiple of 6 times must be divisible by 7.

Now, in the given question digit 6 is repeated 660 times hence it must be divisible by 7, so remainder = 0

**Question 3. **

a) 25

b) 0

c) 50

d) None

### Answer and Explanation

**Solution: option a**