I really need help. It's about number theory. Would yo be so kind help me?

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2 Answers
Mar 21, 2018

Use the fact that if d|ndn, then d|n^2dn2 and if (d|ndn and d|mdm), then d|n-mdnm

Explanation:

Suppose that p=1p=1 or pp is prime, and that p|a+bpa+b and p|a^2-ab+b^2pa2ab+b2

From p|a+bpa+b , we conclude that p|a^2+2ab+b^2pa2+2ab+b2

Therefore, p|3abp3ab, so p|3p3 or p|apa or p|bpb.

Now show that if pp divides aa pr bb, then p = 1p=1

Finish by showing that 9 cancel(|)(a+b,a^2-ab+b^2)

Mar 22, 2018

See below.

Explanation:

Note that

gcd(a+b,a^2-ab+b^2) = gcd((a+b)^2, a^2-a b + b^2)

and also

(a+b)^2 equiv (a²-a b + b^2) mod 3 because

(a²-a b + b^2)-(a+b)^2 = 3 a b

hence

gcd(a+b,a^2-ab+b^2) = {(1),(3):}