# Remainders

Dividend = Quotient Ã— Divisor + Remainder

# Basics of Remainder

- When any positive number A is divided by any other positive number B, and if B > A, then the remainder will be A itself.
- Remainder should always be calculated in its actual form i.e., you can not reduce the fraction to its lower ratio.

Example-1

What is the remainder when 5 Ã— 10

^{5}is divided by 6 Ã— 10^{6}?Solution

As we know that we cannot reduce the fractions to its lower terms, the remainder obtained will be equal to 5 Ã— 10

^{5}.- Concept of negative remainder: As obvious from the name, remainder implies that something has been left out. So, remainder simply can never be negative. Its minimum value can be zero only and not negative.

Example-2

What is the remainder when 50 is divided by 7?

Solution

âˆ’50/7 = (âˆ’56 + 6)/7; which gives a remainder of 6. Or, when we divide â€“50 by 7, we get â€“1 as the remainder. Now, since remainder has to be non-negative, so we add 7 (quotient) to it which makes final remainder as â†’ â€“1 + 7 = 6.

There are two methods to find out the remainder of any expression:

- Cyclicity Method
- Theorem Method

# Cyclicity method

For every expression of a remainder, there comes attached a specific cyclicity of remainders.Example-1

What is the remainder when 4

^{1000}is divided by 7?Solution

To find the cyclicity, we keep finding the remainders until some remainder repeats itself. It can be understood with the following example:
Number/7 â†’ 4
Remainder â†’ 4 2 1 4 2 1 4 2
Now, 4

^{1 }4^{2 }4^{3 }4^{4 }4^{5 }4^{6 }4^{7}4^{8}^{4 }gives us the same remainder as 4

^{1}, so the cyclicity is of 3 (Because remainders start repeating themselves after 4

^{3}.)

^{999}will give 1 as the remainder.

Example-2

What is the remainder when 4

^{96}is divided by 6?**(CAT 2003)**Solution

Finding out the cyclicity,

Number/6 â†’ 4

^{1 }4^{2 }4^{3 }4^{4 }4^{5 }4^{6 }4^{7}4^{8}Remainder â†’ 4 4 4 4 4 4 4 4
The remainder in all the cases is 4, so the final remainder will be 4. Actually, there is no need to find remainders upto 4

^{8}or even 4

^{3}. 4

^{2 }itself gives us a remainder of 4 when divided by 6, which is the same as the remainder obtained when 4

^{1 }is divided by 6. So, we have got the cyclicity here, which is of 1.

^{100}/6 = 2

^{200}/6 = 2

^{199}/3, then remainder obtained will be 2, which is not the right answer (as given in CAT brochure of next year i.e., CAT 2004)

# Remainder Theorem

The product of any two or more than two natural numbers has the same remainder when divided by any natural numbers, as the product of their remainders.Let us understand this through an example.

Example-1

Remainder= Remainder = 2

Solution

Normal way of doing this is â€“ Product â†’â†’â†’ Remainder

Theorem method â€“ Remainder â†’â†’â†’ Product â†’â†’â†’ Remainder

So, first of all we will find the remainders of each number individually and then multiply these individual remainders to find out the final remainder.

Remainder 12/7 = 5

Remainder 13/7 = 6

Remainder = Remainder (5 Ã— 6)/7

= Remainder 30/7 = 2

= Remainder 30/7 = 2

Example-2

What is the remainder obtained when (1421 Ã— 1423 Ã— 1425) is divided by 12?

**(CAT 2000)**Solution

Remainder of 1421/12 = 5

Remainder of 1423/12 = 7

Remainder of 1425/12 = 9

Remainder (1421 Ã— 1423 Ã— 1425)/12

= Remainder (5 Ã— 7 Ã— 9)/12

= Remainder (5 Ã— 63)/12

= Remainder (5 Ã— 3)/12 = 3

Remainder of 1423/12 = 7

Remainder of 1425/12 = 9

Remainder (1421 Ã— 1423 Ã— 1425)/12

= Remainder (5 Ã— 7 Ã— 9)/12

= Remainder (5 Ã— 63)/12

= Remainder (5 Ã— 3)/12 = 3