When proving several statements are equivalent, we need to be able to get from any one statement to any other.
by definition of congruence.
by definition of mod as the remainder.
Assume
for some
By the Quotient-Remainder Theorem,
with
Thus,
Solving for
we get
But by uniqueness of the remainder,
has remainder
Now assume
and
have the same remainder when divided by
By the Quotient-Remainder Theorem,
and
Substituting in for
Since
let