If n = 3� - 2�, which of the following is NOT a factor of n?
(A) 97
(B) 65
(C) 35
(D) 13
(E) 5
Here's a way to determine the correct answer without recognizing that 3� - 2� is the difference of two squares.
Since 65 = 5*13, answer choices B, D and E cancel each other out:
If 5 is not a factor of n, then neither is 65.
If 13 is not a factor of n, then neither is 65.
If 65 is not a factor of n, then 5 is not a factor of n, 13 is not a factor of n, or neither 5 nor 13 is a factor of n.
Each of these cases implies that B and at least one other answer choice is correct.
Since it's not possible that more than one answer choice is correct, eliminate B, D and E.
Thus, either 97 or 35 is not a factor of n.
Since 35 = 5*7, check whether 5 and 7 divide into n.
Since 3� = 81, 3� = 81*81, which can be calculated relatively quickly:
81 * 81 = 6561.
Every test-taker should know the powers of 2 up to 2¹�.
Since 2� = 256, we get:
3� - 2� = 6561 - 256 = 6305.
Since 7 is a factor of 6300, it cannot be a factor of 6305.
After 6300, the next greatest multiple of 7 = 6300+7 = 6307.
Thus, neither 7 nor 35 is a factor of 6305.
The correct answer is
C.
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