Question

Letaandbbe integers andda positive integer.(a) Prove that ifddividesaandddividesb, thenddivides botha+banda−b.(b) Is the converse of the above true? If so, prove it. If not, give a specific example ofa, b, dshowingthat the converse is false.

Answer 1

See step by step explanation.

Explanation:

Recall that given two integers a, b, a divides b if there exists an integer k such that b = ka.

Let a,b,d be integers, such that d>0.

a) Suppose that d divides a and d divides b. Then, there exists
`k_1,k_2 \in \mathbb{Z}` such that
`a = k_1 d` and
`b = k_2 d`. Consider a+b and a-b. Replacing the previous equation, we have that

`a+b = k_1 d + k_2 d = (k_1+k_2) d`

`a-b = k_1 d - k_2 d = (k_1-k_2) d`

Since
`k_1,k_2\in \mathbb{Z}` then
`k_1+k_2` and
`k_1-k_2` are both integers. Then, d divides both a+b and a-b.

b) It is false. Let a = 7, b = 5. Then d = 2 divides a+b (12) and a-b (2) but neither 2 divides 7 nor 2 divides 5.