NandishSS wrote: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
8! = 8*7*6*5*4*3*2 = (2³)(7)(2*3)(5)(2²)(3)(2) = (2�)(3)(5)(7).
Since the product above is a multiple of a^n, we get:
(2�)(3)(5)(7) = (a^n)(k), where k is a positive integer.
Statement 1:
It's possible that a=2, n=6 and k=2*3*5*7, with the result that the following values in red are equal:
(2�)(3)(5)(7) =
(2�)(2)(3)(5)(7)
It's possible that a=4, n=3 and k=2*3*5*7, with the result that the following values in blue are equal:
(2�)(3)(5)(7) =
(4³)(2)(3)(5)(7).
Since a can be different values, INSUFFICIENT.
Statement 2:
(2�)(3)(5)(7) = (a�)(k).
The equation above is valid only if a=2 and k=2*3*5*7, as follows:
(2�)(3)(5)(7) =
(2�)(2)(3)(5)(7).
Thus, a=2.
SUFFICIENT.
The correct answer is
B.
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