Question: Which one of the following is not true in relation with (, ), the g.c.d. of two numbers.
(a) If (a,b) = (a,c) = 1, then (a,bc) = 1
(b) (2,n) = 1 for any odd integer n
(c) (a, 1) = (1, a) = 1
(d) (a, 0) = (0, a) = 0
Solution:
(a) (a,b) = 1
ax+by = 1------(1)
(a,c) = 1
au+cv = 1-------(2)
(1)(2)---(ax+by)(au+cv) = 1
a(axu+xcv+byu)+bcyv = 1
(a,bc) = 1
(b) N and 2 have no common divisor if N is an odd number. So (2, N) = 1
(c) It is clear that (1, a) = (a, 1) = 1
(d) gcd(a, 0) = gcd(0, a) = |a|
So the correct answer is (d)
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