Another GMAT Prep Question

[mindruna]

If the integers a and n are greater than 1 and the product of the first 8 integers is a multiple of a^n, what is the value of a?

1) a^n=64
2) n=6


The OA is b, but I'm not certain how that is sufficient. Any insight would be really appreciated! Thanks.

[gabriel]

If the integers a and n are greater than 1 and the product of the first 8 integers is a multiple of a^n, what is the value of a?

1) a^n=64
2) n=6


The OA is b, but I'm not certain how that is sufficient. Any insight would be really appreciated! Thanks.

the question seems incomplete to me ... what is the darkened part supposed to mean .. which integers are we talking about .. plz take a look at the question again and repost ..

[blah45]

If the integers a and n are greater than 1 and the product of the first 8 integers is a multiple of a^n, what is the value of a?

1) a^n=64
2) n=6


The OA is b, but I'm not certain how that is sufficient. Any insight would be really appreciated! Thanks.

product of the first 8 integers is a multiple of a^n
so, ( 1*2*3*...*8 )/(a^n) = k => where a, n, and k are all integers.

(1)
4^3 = 64 => a=4, n=3
8^2 = 64 => a=8, n=2
2^6 = 64 => a=2, n=6
INSUFF
(2)
n=6 => only a=2 can satisfy this.
SUFF

[gabriel]

If the integers a and n are greater than 1 and the product of the first 8 integers is a multiple of a^n, what is the value of a?

1) a^n=64
2) n=6


The OA is b, but I'm not certain how that is sufficient. Any insight would be really appreciated! Thanks.

product of the first 8 integers is a multiple of a^n
so, ( 1*2*3*...*8 )/(a^n) = k => where a, n, and k are all integers.

(1)
4^3 = 64 => a=4, n=3
8^2 = 64 => a=8, n=2
2^6 = 64 => a=2, n=6
INSUFF
(2)
n=6 => only a=2 can satisfy this.
SUFF

Well, the above means the answer should be [D], not . We are looking to justify . Good thinking though. Thanks

actually that means the answer is C .. anyway, that does not matter bcoz u cant reach the conclusion in blah45's solution without knowing which "first 8 integers" is supposed to be meant over here ...

[bingojohn]

Well, the above means the answer should be [D], not . We are looking to justify . Good thinking though. Thanks

How is it (D)? I mean it's clear from statement (1) 'a' could have three possible values. ONLY statement (2) indicates that a=2 and n=6 work.

I am sorry, my bad. The answer would be , in fact. The original answer is , which is correct. Here is how:

product of first 8 integers:
8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

taking (2) into consideration, n = 6... then prime factorize the left hand side:

(2^3) x 7 x (2 x 3) x 5 x (2 x 2) x 3 x 2 x 1 = (a^6) k
(2^7) x (3^2) x 5 x 7 = (a^6) k

The only value of 'a' that satisfies the above is: a = 2

Hence (2) alone is SUFFICIENT... (1) alone is NOT SUFFICIENT...