let us assume two numbers A and B. Now A is not divisible by B.

mod = A%B

now (A + mod*(B-1)) is divisible by B. Why???

for example

A = 3

B=5

-> mod = A%B = 3%5 = 3;

-> A + mod*(B-1) = 3 + 3*(5-1) = 3 + 12 = 15 which is divisible by B.

another example:

A = 17

B = 12

mod = 17%12 = 5

A + mod * (B-1) = 17 + 5*(12-1) = 17 + 5*11 = 72 which is divisible by B = 12.

Try for any example and this is true. But i want to know why.?