Bunuel wrote:
\(x = 9^{10}– 3^{17}\) and x/n is an integer. If n is a positive integer that has exactly two factors, how many different values for n are possible?
(A) One
(B) Two
(C) Three
(D) Four
(E) Five
Since n has exactly two factors, n must be prime. Thus, we should prime factorize x:
x = (3^2)^10 - 3^17 = 3^20 - 3^17 = 3^17(3^3 - 1) = 3^17(26) = 3^17(2)(13)
Thus, n could be 3, 2, or 13.
Answer: C
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