Question

if x is a three digits number and y is a number obtained by Purmuting the digits of x is any manner then x-y is always divisible by

(a) 4
(b) 6
(c) 9
(d) 12​

Answer 1

(c) 9

Given,

x is a three-digit number.

To find,

If the digits are interchanged, the difference of x - y is always divisible by?

Solution,

x is the given three-digit number.

It can be represented in the form of a*100 + b*10 + c.

Where a is the digit at hundred place, b is at tens place, and c is at one place.

Now y is another three-digit number where b is the digit at hundred place, c is at tens place, and a is at one place.

It can be represented in the form of b*100 + c*10 + a.

x - y = a*100 + b*10 + c - ( b*100 + c*10 + a )

x - y = a*100 + b*10 + c - b*100 - c*10 - a

x - y = a*99 - b*90 - c*9

x - y = 9 ( 11*a - 10*b - c )

We can see that 9 comes out as one of the factors for x - y.

Therefore, the difference x - y is always divisible by 9.