
64. Let a, b, and c be three counting numbers. Set d = GCD(ac, bc) and e = GCD(a, b).
We want to prove that d = ce.
Part 1. d ≥ ce
Because e = GCD(a, b), we can write a = ke and b = se with k an d s relatively prime. Multiplying
both equalities by c, we obtain
ac = kðceÞ and bc = sðceÞ:
This proves that ce is a common divisor of ac and bc.Butd is the greatest common divisor.
Thus, d ≥ ce.
Part 2. d ≤ ce
As e is the greatest common divisor of a and b, we can write a = ke and b = se ,w
ithk and s
relatively prime. So, multiplying by c, we obtain
ac = kðceÞ and bc = sðceÞ,
with k and s relatively prime. Then all the common factors of ac and bc are in ce.T