# Set theory: Problem 4

### Hints

Do you understand the set-builder definition of L(s,t)?
L(s,t) consists of all integers that can
be expressed as the sum of multiples of s and t.
If you can't form a picture of what this might mean, pick
some concrete values for s and t. Calculate some values
that live in L(s,t). Suppose s=6 and t=9. Find some
integers that cannot be written in the form sx+ty (where
you get to pick the integers x and y).

You need to prove
\(L(a,m) \subseteq L(r,m)\).
So pick an element x in L(a,m).
You need to show that x is in L(r,m).
Remember that L(a,m) and L(r,m) contain integers, so x will be an integer.

You've been asked to use the textbook definitions of congruence
and divisibility. Look them up if you don't remember which
form was our official definition.