Data Sufficiency - 38

If q is a integer, is q^4 a multiple of 64?

(1) q^4 is not a multiple of 128.
(2) q^2 has 27 factors, 7 of which are less than or equal to 10

Answer: A

OE:
From statement (1): Given q is an integer, thus q can be written as product of distinct prime factors, where the power of 2 must be a whole number(a non-negative integer). This implies that the power of 2 in the prime factorization of q^4 must be a multiple of 4. As 2^7 is not a factor of q^4, the highest power of 2 that could be a factor of q^4 is 2^4.
Hence q^4 is not a multiple of 64=2^6......sufficient

From statement (2): Given that q^2 has 27 factors. Thus q^2 could be of the form
(i) a^26, or
(ii)b^2*c^8 or
(iii) a^2*b^2*c^2 where a b and c are distinct prime numbers.

Because q^2 has 7 factors less than 11, (i) is impossible.

As for (ii) 2^8*3^2 has exactly 7 factors under 11 (1,2,3,4,6,8,9), in which case q^4 would be a multiple of 64. 3^8*2*2 is not a possibility for q^2 (it only has 6 factors under 1: 1,2,3,4,6,9)

Regarding (iii) if q^2=2^2*3^2*5^2, q^2 would have 8 factors under 11- 1,2,3,4,5,6,9,10 and q^2=2^2*3^2*7^2 would have 7 factors under 11: 1,2,3,4,6,7,9) In this case, q^4 would not be a multiple of 64. Thus (2) is insufficient

hence the answer A