Does this proof seem valid?
Can someone please guide me? I have a find exam tomorrow morning and I am trying to make sure I understand this.
Prove: For all integers a,b, and c, if a|b and a|c then a|(b+c).
Suppose that a,b, and c are any integers where a|b and a|c.
Let b=a*k for some integer k (by definition of divisibility).
Let c=a*p for some integer p (by definition of divisibility).
b+c= ak+ap (by substitution).
Hence, ak+ap is an integer b/c integers and sums and products of integers are integers.
Therefore, a|(b+c)= b+c=a*z for some integer z.
So, ak+ap= az for some integer z (by substitution)
Solve for z to prove that z is an integer:
Therefore, a is divisible by b+c since b+c= a*z for some integer z.
**END OF PROOF**.