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

(1) a^n = 64
\(2^6 , 4^3 , 64^1 , 8^2\) - we get multiple answers, so Option I - NOT SUFFICIENT

(2) n = 6
We do not know the value of a, Option II - NOT SUFFICIENT

(1) and (2) together
n=6 and \(a^n\)=64, so we know that a = 2
(1) and (2) tother - SUFFICIENT

Ans C
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so is the given information "the product of the first 8 positive integers is a multiple of a^n" irrelevant?
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Bunuel
If the integers a and n are positive and the product of the first 8 positive integers is a multiple of a^n, what is the value of a ?

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


Similar question from GMAT Prep: https://gmatclub.com/forum/if-the-integ ... 32382.html

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(a^n) x k = 1x2x3x4x5x6x7x8
(a^n) x k = (1)x(2^6)x(3^2)x(5)x(7).

(1) a^n = 64
2^6 = 64
(2^5)x2 = 64
4^3 = 64
(4^2)x4 = 64
8^2 = 64

'a' can have many possible values.
Therefore statement (1) alone is insufficient.

(2) n = 6
(a^n) x k = (1)x(2^6)x(3^2)x(5)x(7).
'a' could be 1 or 2.
Therefore statement (2) alone is insufficient.

Combining (1) and (2) together.
'a' = 2

Answer C - Statement (1) and (2) together are sufficient.
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so is the given information "the product of the first 8 positive integers is a multiple of a^n" irrelevant?

I think that information is a trap because if you don't consider a=1 (1^6) as a quotient of 8! {1^6 x 2^7 x 3^2 x 4 x 5 x 7) you can end concluding that b alone is sufficient to answer the question.
Notice that 8! only have two values with exponents where n can equal 6 (1^6 and 2^7).

Hope this helps.
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