If the integers a and n are greater than 1 and the product : DS Archive
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# If the integers a and n are greater than 1 and the product

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Manager
Joined: 04 Nov 2006
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If the integers a and n are greater than 1 and the product [#permalink]

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11 Nov 2006, 15:54
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If the integers a and n are greater than 1 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
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11 Nov 2006, 16:12
(1) => 8^2,4^3,2^6 Insuff.
(2) => a = 2 because 2^6 is the only number that can divide 1*2*3*4*5*6*7*8 evenly. Suffcient.

B.

I almost said C when I first looked at the problem
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11 Nov 2006, 16:21
G.Matter,

How did you determine that 2^6 is the only number that can divide 1*2*3*4*5*6*7*8 evenly. Is there a trick to knowing this?

Artis
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11 Nov 2006, 16:27
Unfortunately I did not employ any trick. As I mentioned, in this case I almost chose C, then what made me stop was I realized that I wasnt using this information anywhere ->
the product of the first 8 positive integers is a multiple of a^n,

Then I just wrote down all the factors and realized 2 is the only possibility. It did take me sometime.
11 Nov 2006, 16:27
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# If the integers a and n are greater than 1 and the product

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