The number of homomorphisms from (Z,+) to (Q*, .) such that f(2) = 1/3.
Let f be such a homomorphism. Then f is completely determined by f(1). (If f(1) = x, then f(n) = xn for any integer n.) Let f(1) = x, a rational. Then x2 = 1/3. Ask yourself how many rationals x have this property. The answer should now be clear.