Prove Forall a, b, c elementof Z^+, if a| (b + c) and a|c then a|b.

Answer with explanation:

If A is a positive Integer, then if A divides B, then in terms of equation it can be written as

→B=A m, where m is any integer.

⇒Now, it is given that, three elements, a, b and c belong to set of Integers.

a divides b+c, and a divides c

then we have to prove that, a divides b.

Proof

→b+c = k a, where k is an integer, as b+c is divisible by a.

→Also, c = m a, where m is an integer. Because c is divisible by a.

→b + m a = k a

→b=k a - ma

→b=a (k - m)

Since, k and m are both integers. So, k-m will be also an integer.

Let, k-m = p

→b=a p

which shows that, b is divisible by a.

Hence proved.