How many positive perfect cubes are divisors of 4^6?

fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 16)

How many positive perfect cubes are divisors of 4^6 ?

(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

Thank you for your nice contribution, Brent!

$$?\,\,\,:\,\,\,\# \,\,{M^3}\,\,\,\left( {M \ge 1\,\,{\mathop{\rm int}} } \right)\,\,\,{\rm{such}}\,\,{\rm{that}}\,\,{{{4^6}} \over {{M^3}}}\,\,\mathop = \limits^{\left( * \right)} \,\,{\mathop{\rm int}} $$
$${4^6} = {2^{12}} = {\left( {{2^4}} \right)^3}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,{\left( {{{{2^4}} \over M}} \right)^3} = {\mathop{\rm int}} \,\,\,\,\, \Rightarrow \,\,\,\,\,M = {2^0},{2^1},{2^2},{2^3},{2^4}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {\rm{D}} \right)$$


The correct answer is (D).

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.