Proving a statement about divisors.

[cragar]

Homework Statement

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

The Attempt at a Solution

Proof:
We wish to show that if a|b and a|c then a|(b+c)
Let a,b and c be integers. If a|b then there exists an integer x such that ax=b,
and if a|c then there exists an integer y such that ay=c. If a|(b+c) then there exists an integer z such that az=(b+c).
we see that b+c=ax+ay=az
a(x+y)=az=b+c
and we see that (b+c) is divisible by a.
Is this good enough?

 

[tiny-tim]

hi cragar!

yes, exactly the right idea, but instead of telling the story from start to finish, you put the end in the middle

better would be …

Let a,b and c be integers. If a|b then there exists an integer x such that ax=b,
and if a|c then there exists an integer y such that ay=c.

[STRIKE]If a|(b+c) then there exists an integer z such that az=(b+c).[/STRIKE]
[STRIKE]we see that b+c=ax+ay=az[/STRIKE]
but then x+y is an integer, and a(x+y) = ax + ay = b+c

so a | b+c ​

don't give away the ending too early!

 

[cragar]

ok, thanks for the advice. I just want to make sure I am constructing my proofs in a correct way.