sjuniv32 wrote:
If p and n are positive integers greater than 2, is \(p^n − p^{n−2}\) divisible by 8?
1) n is odd, and n > 5
2) p is odd, and p > 5
We can factor out \(p^{n−2}\) and ask if \(p^{n -2}*(p^2 - 1) = p^{n-2}*(p - 1)(p + 1)\) is divisible by 8.
Statement 1:Since n > 5, we have \(n - 2 > 3\). Then the power of p is at least 4.
If p is even, \(p^{n - 2}\) will have at least 4 multiples of 2, then the entire product would be a multiple of 8.
If p is odd, then \((p - 1)*(p + 1) = \text{Even * Even}\). Note these two evens are consecutive, so we must have a multiple of 4 and a multiple of 2 multiplied together. Then the product must have a multiple of 8.
Sufficient.
Statement 2:Same as the odd case in statement 1. Sufficient.
Ans: D
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