Confirm the following properties of the greatest common divisor:
(a) If gcd(a, b) = 1, and gcd(a, c) = 1, then gcd(a, bc) = 1.
[Hint: Because 1 = ax + by = au + cv for some x, y, u,v,
1 = (ax + by)(au + cv) = a(aux + cvx + byu) + bc(yv).]
(b) If gcd(a, b) = 1, and c | a, then gcd(b, c) = 1.
(c) If gcd(a, b)= 1, then gcd(ac, b) = gcd(c, b).
(d) If gcd(a, b) = 1, and c | a + b, then gcd(a, c) = gcd(b, c) = 1.
[Hint: Let d = gcd(a, c). Then d|a,d|c implies that d | (a + b) − a, or d | b.]
(e) If gcd(a, b) = 1, d | ac, and d | bc, then d | c.
(f) If gcd(a, b) = 1, then gcd(a2, b2) = 1.
[Hint: First show that gcd(a, b2) = gcd(a2, b) = l.]
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