Number System - 4

Remainder Rules

Rule 1

When the same number is dividend by the two different divisors such that one divisor is a multiple of the other divisor and also the remainder for the greater divisor is known.

If the remainder for the greater divisor = r

And the smaller divisor = d, then

That when r > d, the required remainder for the smaller divisor will be the remainder found out by dividing the ‘r’ by ‘d’.

And when r < d, then the required remainder is ‘r’ it self.

Ex.:- If a number is divided by 527, the remainder is 42. What will be the remainder if it is divided by 17?

Soln.:- Here the same number is divided by two divisors: 527 and 17.

Now, 527/17 = 31, so, 527 is a multiple of 17.

Remainder of the greater divisor = 42 and smaller divisor = 17.

So, 42/17 = 8 = required remainder for the smaller divisor.

Hence, I f the number is divided by 17, the remainder will be 8.

Rule 2

If two different number a and b, on being divided by the same divisor leave remainder ‘r1’ and ‘r2’ respectively, then their sum (a + b) if divided by same divisor will leave remainder R, given by

R = (r1 + r2) – divisor

The required remainder R = sum of remainders – divisor (when some is divided)

Note:- If ‘R’ becomes negative in the above equation, then the required remainder will be the sum of the remainder.

The required remainder = sum of remainders.

Ex.:- Two different numbers, when divided by the same divisor, leave remainders 15 and 39 respectively, and when their sum was divided by the same divisor, the remainder was 7. What is the divisor?

Soln.:- Using the rule 2

7 = (15 + 39) – divisor

Divisor = 47.

Rule 3

When two numbers, after being divided by the same divisor leave the same remainder, then the different of those two numbers must be exactly divisible by the same divisor.

Ex.:- Two numbers 147 and 225, after being divided by a 2-digit number, leave the same remainder. Find the divisor.

Soln.:- By rule 3, the difference of 225 and 147 must be perfectly divisible by the divisor.

The difference = 225 – 147 = 78

Now, 78 = 13 x 2 x 3

Thus, 1-digit divisors = 2, 3 and 2 x 3 = 6

2-digit divisors = 13, 13 x 2 = 26, 13 x 3 = 39, 13 x 2 x 3 = 78

So, the possible divisors are 13, 26, 39, 78.

Rule 4

If a given number is divided successively by the different factors of the divisor leaving remainder ‘r1’, ‘r2’, and ‘r3’ respectively, then the true remainder (i.e. remainder when the number is divided by the divisor) can be obtained by using the following formula:

True remainder = (first remainder) + (second remainder x first divisor) + (third remainder x second divisor x first divisor)

Ex.:- A number, being successively divided by 5, 7 and 11 leaves 3, 1, 2 as remainders respectively. Find the remainder if the same number is divided by 385.

Soln.:- Here, the divisor is 385, whose factors are 5, 7 and 11

By the rule 3, True remainder (i.e. remainder when divided by 385) = 3 + (1 x 5) + (2 x 5 x 7)

= 3 + 5 + 70 = 78

Rule 5

When (x + 1)ⁿ is divided by x, the remainder is always 1, where x and n are natural numbers.

Ex.:- What will be the remainder when (17)²¹ is divided by 16?

Soln.:- (17)²¹ = (16 + 1)²¹

When (16 + 1)²¹ is divided by 16, the remainder = 1.

Rule 6

When (x – 1)n is divided by x, then

The remainder = 1, when n is an even natural number

But the remainder = x-1, when n is an odd natural number

Ex.:- What will be the remainder when (29)⁷⁵ is divided by 30?

Soln.:- (29)⁷⁵ = (30 – 1)⁷⁵, here index = 75 (which is odd) so, when (30 – 1)⁷⁵ is divided by 30, the remainder will be x – 1 = 30 – 1 = 29.