This is for a abstract algebra course. Let a, b, c be integers. Give a proof for

Question

This is for a abstract algebra course.

Let a, b, c be integers. Give a proof for these facts about divisors:

(a) If b|a and c|b, then c|a.

(b) If b|a and b|(a+c) then b|c.

I have an example and some background information that might help:


"a|b" means that a divides b; it also means that b=qa

Ex.-
If a|b and c is an integer, then a|bc.  Proof: Since a|b, there is an integer q such that b = qa.  So cb = cqa.  We need some integer p such that bc = pa.  Looking at cb = cqa shows that letting p = cq works.

Ex.-

If a|b and a|c then a|(nb+mc) for all integers n and m. Proof: By the previous fact, we have integers q1 and q2 such that nb = aq1 and mc = aq2.  So nb+mc = aq1+aq2 = a(q1+q2).

Explanation / Answer

a) b|a => a= kb for some integer k and c|b => b= cl for some integer l => a=kb= k(cl) = (kl) c => c|a b) b|a => a= kb for some integer k and b|(a+c)=> a+c = lb for some integer l kb+c=lb => c=b(l-k) (l-k is an integer) so, b|c

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