Problem

Assuming that gcd(a, b) = 1, prove the following:(a) gcd(a + b, a − b) = 1 or 2.[Hint: Let...

Assuming that gcd(a, b) = 1, prove the following:

(a) gcd(a + b, ab) = 1 or 2.

[Hint: Let d = gcd(a + b, ab) and show that d|2a, d|2b, and thus that d ≤ gcd(2a, 2b) = 2 gcd(a, b).]


(b) gcd(2a + b, a + 2b) − 1 or 3.


(c) gcd(a + b, a2 + b2) = 1 or 2.

[Hint: a2 + b2 = (a + b)(ab) + 2b2.]


(d) gcd(a + b, a2ab + b2) = 1 or 3.

[Hint: a2ab + b2 = (a + b)2 − 3ab.]

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Solutions For Problems in Chapter 2.4