/sci/ - Science & Math

How's my proof?
a, b, x and y are positive integers. Prove that if a is congruent to b modulo 5, and x is congruent to y modulo 5, then ax is congruent to by modulo 5.
Proof:
>Suppose a is congruent to b modulo 5 and x is congruent to y modulo 5.
>Then a - b = 5c and x - y = 5d, where c and d are positive integers
>ax - by = 5e, where e is a positive integer
>a = b + 5c
>x = y + 5d
>(b + 5c)(y + 5d) - by = by + 5bd + 5cy + 25cd - by = 5(bd + cy + 5cd)
>Then, ax - by = 5k, where k is bd + cy + 5cd.
>Therefore, ax is congruent to by modulo 5.