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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.

Respuesta :

Answer:

See step by step explanation.

Step-by-step 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 [tex]k_1,k_2 \in \mathbb{Z}[/tex] such that [tex]a = k_1 d [/tex] and [tex]b = k_2 d[/tex]. Consider a+b and a-b. Replacing the previous equation, we have that

[tex]a+b = k_1 d + k_2 d = (k_1+k_2) d [/tex]

[tex]a-b = k_1 d - k_2 d = (k_1-k_2) d[/tex]

Since [tex]k_1,k_2\in \mathbb{Z}[/tex] then [tex]k_1+k_2[/tex] and [tex]k_1-k_2[/tex] 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.

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