$\displaystyle a,b,c,n,m \in \mathbb{Z}$

If a|b and a|c, show that a|(bm+cn).

Proof:

I have already proved that if x|y, and y|z, then x|z.

So, if a|b, then b|bm, so a|bm.

Also, if a|c, then c|cn, so a|cn.

So I think I need to show that a|aj+ai for some integers j=bm, i=cn.

ax = aj+ai

ax = a(j+i)

Let x = (j+i), thus ax = ax. $\displaystyle \blacksquare$

Could someone let me know if this proof is correct?