2010/002) The numbers 12n + 1 and 30n + 2 are relatively prime for ...?

any positive integer n;
b) infinitely many, but not all, integers n;
c) for infinitely many integers n;
d) none of the above;
Kindly explain...

Ans:
GCD(12n + 1 ,30n + 2)
= GCD(12n + 1 ,2(15n+1))
= GCD(12n + 1 ,15n+1) as 12n+1 is odd
= GCD(12n+1,(15n+1-(12n+1)) as GCD(a,b) = GCD(a,b-a)
= GCD(12n+1,3n)
= GCD(3n, 12n+1- 4*3n) as GCD(a,b) = GCD(b,a- n * b))
= GCD(3n,1)
= 1 for all n
hence ans is a