Remainders: Part-5
In this article we will discuss calculating remainders that involve hefty mathematical expressions. We will discuss each and every mathematical expression one by one.

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

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.
Let’s see application of above in problems

Example
What will be the remainder when
(576 + 453 -34 – 566 +6734 x 234 x 345 x 4564) is divided by 8 ?
Solution:
So to find the remainder we will not solve all the expression we will divide it one by one separately and apply same operations on remainders as we have in the numbers.
First divide 576 by 8 and remainder is 0
Then divide 453 by 8 and remainder is 5
Then divide 34 by 8 and remainder is 2
Then divide 566 by 8 and remainder is 6
Then divide 6734 by 8 and remainder is 6
Then divide 234 by 6 and remainder is 2
Then divide 345 by 8 and remainder is 1
Then divide 4564 by 8 and remainder is 4
Now apply same operations between remainders and solve
0 + 5 -2 -6 +6 x 2 x 1 x 4
= 45
Now 45 is greater than the 8 so we will again divide the result with 8.
So divide 45 by 8 gives remainder 5 so the remainder of the whole expression is 5.