Discrete Math Proof

[intervade]

Alright, Im sort of understanding these but I think Im missing like one step really, just need someone to point me in the right direction. Here are some of them:

Suppose a, b and c are integers. Prove that if a|b and a|c then a|(b+c)
Also, Suppose a, b and c are integers. Prove that if a|b then a|(bc)

Prove that if a|b and a|c then a|(b+c):

Basically I started out defining a|b and a|c with our basic definition of division. There is an integer x such that b = ax and there is an integer z such that c = az. I also said that there is an integer y such that (b+c) = ay. But I'm last as to where to from here.

I also took the same approach with the second problem. But obviously bc = ay.

Any help would be much appreciated!

 

[daon]

You;re on the right track. If b=ax and c=az then b+c = ax+az. By the distributive property for multiplication over division, we have ax+az=a(x+z). Hence (b+c) = a(x+z). Since the integers are closed under addition x+z is an integer. Therefore, ... <finish this>

You said ," that there is an integer y such that (b+c) = ay. " NO. That is what you are trying to prove.

The second one is easier: a|b => b=ax => bc = (ax)c = a(xc)