looks fine to me
Can somebody please check my work on this proof? I would greatly appreciate it!
Prove a ≡ b(mod m) if and only if there exists an integer k such that a = km+b.
(1)
If a ≡ b(mod m), then a = km+b, where k is an existing integer.
Proof: (⇒)Suppose a ≡ b(mod m), then m|a-b.
Thus a-b = km where k is an existing integer.
Therefore, by addition, a = km+b.(⇐)
(2)
If a =km+b and k is an existing integer, then a ≡ b(mod m).
Proof: Suppose a=km+b for an existing integer k.
Then a-b=km. Thus m|a-b.
Therefore a ≡ b(mod m).
Hence, a ≡ b(mod m) ⇐⇒ a=km+b for an existing integer k.