To find the "factor count" of an integer, you'll need the prime factorization, or something close to it. Consider each choice:
(A) 60 = 30(2) = 15(2)(2) = 3(5)(2)(2) -- three distinct prime factors.
(B) 61 is prime -- one distinct prime factor.
(C) 62 = 31(2) -- two distinct prime factors.
(D) 63 = 9(7) = 3(3)(7) -- two distinct prime factors.
(E) 64 = 2^6 -- one distinct prime factor.

(A) is correct.