Number System - 3

Division and Remainder

When a given number is not exactly divisible by any number, then there is a remainder number at the end of such division.

Suppose we divide 25 by 7 as,

Then, divisor =7, dividend = 25, quotient = 3 and remainder = 4.

There is most important formula to check the divisibility:-

Dividend = (divisor x quotient) + remainder

So, if a number ‘x’ is divided by ‘k’, leaving remainder ‘r’ and giving quotient ‘q’ then the number can be found by using (i)

x = kq + r

Hence, if the number x is exactly divisible by k, then remainder = r = 0

x = kq

and so ^x/_k = q, implying that x is divisible exactly by ‘k’ and ‘q’ is an integer.

Methods to Find a Number Completely Divisible by Another

Consider a given number x. when divided by ‘d’, it gives a quotient ‘q’ and remainder ‘r’.

Now, to find a number exactly divisible by ‘d’, we can use either of the following two methods to reduce the remainder to zero. ( If a number is exactly divisible, then remainder is zero).

Method 1

By subtracting the remainder from the given number ( dividend).

The required number that is exactly divisible by ‘d’ = x – r

Hence ‘remainder’ is the least number that can be subtracted from any number to make it exactly divisible.

Method 2

By adding the (divisor – remainder) to the given number.

The required number that is exactly divisible by d = x + (divisor – remainder)

Therefore, ( divisor – remainder ) is the least number that can be added to any given number to make it exactly divisible.

Example:

Find the least number, that must be

(a) Subtract from, or

(b) Added to a given number 5029, to make it exactly divisible by 17.

Soln.:- On dividing 5029 by 17, we get remainder = 14.

(a) The least number to be subtracted to make it exactly divisible = remainder = 14.

(b) The least number to be added to make it exactly divisible = divisor – remainder = 17 – 14 = 3.

Greatest n-digit and Least n-digit number Exactly Divisible by a number

(a) To find out the greatest n –digit number exactly divisible by a divisor ‘d’, we use method 1.

The required number = greatest n-digit number – remainder.

(b) To find out the least n-digit number exactly divisible by a divisor ‘d’, we use method 2, because if we use method 1, then subtracting any number from the n-digit least number will reduce it to (n-1) digit number.

The required number = least n-digit number + (divisor – remainder)

Example:- Find the

(a) greatest 3-digit number divisible by 13.

(b) The least 3-digit number divisible by 13.

Soln.:-

(a) Greatest 3-digit “999” divided by 13, we get remainder = 11

So, the required 3-digit greates number = 999 – 11 = 988

(b) Least 3-digit “100”divided by 13, we get remainder = 9

So, the required 3-digit least number = 100 + (13 – 9) = 104.