Loading....
Coupon Accepted Successfully!

 

Remainders

Dividend = Quotient × Divisor + Remainder

Basics of Remainder

  1. When any positive number A is divided by any other positive number B, and if B > A, then the remainder will be A itself.
     
    For example remainder of 5/12 = 5
     
    Remainder of 212/678 = 212
  2. Remainder should always be calculated in its actual form i.e., you can not reduce the fraction to its lower ratio.
     
    For example, remainder of 1/2 = 1
     
    Remainder of 2/4 = 2
     
    Remainder of 3/6 = 3
     
    It can be observed that despite all the fractions being equal, remainders are different in each case.
Example-1
What is the remainder when 5 × 105 is divided by 6 × 106?
Solution
As we know that we cannot reduce the fractions to its lower terms, the remainder obtained will be equal to 5 × 105.
 
  1. 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
−50/7 = (−56 + 6)/7; which gives a remainder of 6. Or, when we divide –50 by 7, we get –1 as the remainder. Now, since remainder has to be non-negative, so we add 7 (quotient) to it which makes final remainder as → –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 41000 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 → 41 42 43 44 45 46 47 48
 
Remainder → 4 2 1 4 2 1 4 2
 
Now, 44 gives us the same remainder as 41, so the cyclicity is of 3 (Because remainders start repeating themselves after 43.)
 
So, any power of 3 or a multiple of 3 will give a remainder of 1. So, 4999 will give 1 as the remainder.
 
Final remainder = 4
 
Example-2
What is the remainder when 496 is divided by 6? (CAT 2003)
Solution
Finding out the cyclicity,
Number/6 → 41 42 43 44 45 46 47 48
Remainder → 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 48 or even 43. 42 itself gives us a remainder of 4 when divided by 6, which is the same as the remainder obtained when 41 is divided by 6. So, we have got the cyclicity here, which is of 1.
 
Hence, final remainder = 4
 
It also can be observed here that if we write 4100/6 = 2200/6 = 2199/3, 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
RemainderDescription: 5710.png= Remainder Description: 5703.png = 2
Solution
Normal way of doing this is – Product →→→ Remainder
 
Theorem method – Remainder →→→ Product →→→ 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 Description: 5697.png = Remainder (5 × 6)/7
= Remainder 30/7 = 2
 
 
Example-2
What is the remainder obtained when (1421 × 1423 × 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 × 1423 × 1425)/12
= Remainder (5 × 7 × 9)/12
= Remainder (5 × 63)/12
= Remainder (5 × 3)/12 = 3
 





Test Your Skills Now!
Take a Quiz now
Reviewer Name