Problem Set Worksheet Page 8
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84. (a) If gcd (a, b) = 1, prove that gcd (a + b, a − b) = 1 or 2.
Proof. Suppose that gcd (a, b) = 1.
Let d = gcd (a + b, a − b) .
This means that d is a common divisor of a + b and a − b, and hence, of their sum,
(a + b) + (a − b) = 2a.
Similarly, d divides the difference of a+b and a−b. (i.e., d divides (a + b)−(a − b) = 2b.)
Since d is a common divisor of 2a and 2b, it follows that
d ≤ gcd (2a, 2b) = 2 gcd (a, b) = 2 · 1 = 2.
i.e., d ≤ 2.
Hence, d = gcd (a + b, a − b) = 1 or 2.
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