What are the possible values of GCF(n, n + 14)?

Question

This is a question for Math for elementary teachers any help would be appreciated.

Answer 1

Assume integer n>0 (strictly positive).

14=2×7, so if n=2p or 7q where p and q are whole numbers (that is, n is even or a multiple of 7), then we have GCF(2p,2p+14)=GCF(2p,2(p+7)) and the GCF is 2 (for p not a multiple of 7) and GCF is 14 (for p a multiple of 7).

Or, we have GCF(7q,7q+14)=GCF(7q,7(q+2)) and the GCF is 7 (for odd q) and GCF is 14 (for even q).

If n is odd and not a multiple of 7, then GCF=1.

Some examples:

n=1, 3, 5, 9, 11, 13, 15, 17, 19, 23, ...: n is odd, GCF=1 (trivial case);

n=2, 4, 6, 8, 10, 12, 16, 18, ...: GCF=2;

n=7, 21, 35, 49, ...: GCF=7;

n=14, 28, 42, 56, ...: n is even, GCF=14.