Question

A lowest number when divided by 32 leaves remainder 30 .what is the remainder when the same is divided by 9?​

Answer 1

Let the number be x. We assume that x is an integer.

Then, we are given that x ~ 21 (mod 32), or, equivalently, x = 32r + 21 for some integer r.

Now, considering x modulo 18, we have that

x = 32r + 21 ~ 14r + 3 ~ -4r + 3 (mod 18).

Letting s := -r, we then have that x ~ 4s + 3 (mod 18).

So x can be congruent to any of 3, 7, 11, 15, 1, 5, 9, 13 or 17 modulo 18. So there are 9 possibilities for the remainder of x after division by 18. This has something to do with 9 being half of 18 and the GCD of 4 and 18 being 2.