Hello could anyone help me prove this Let r s t and w be int

Hello, could anyone help me prove this:

Let r, s, t and w be integers with w > s > 1. If r t (mod w) and s divides w, then r t (mod s).

Aside from stating the assumtions I don\'t have a clue what to do next.

Solution

Given that r t (mod w)

i.e. r when divided by w gives remainder t. (t<w)

r = wq+t where q is the quotient (integer)

Also s divides w.

s = wp where p is another integer

Hence r = wq+t

There are three possibilities for p and q. If p = q

then r = wp+t = s+t

Or r = t mod S is proved.

Case II:

p > q

r = wq+t

= wp+t-w(p-q)t

= s+w(p-q)t where p-q is an integer.

This proves that r = t mod S

Similar proof can be done for q>p also taking q-p as positive integer.

Hence proved