Let's take each statement one by one.AAPL wrote:Manhattan Prep
If \(x, n\), and \(y\) are all positive integers, is \(x^n\) divisible by \(y\)?
1) \(x\) is divisible by \(y^n\).
2) \(x^y\) is divisible by \(y\).
OA A
1) \(x\) is divisible by \(y^n\).
Say x = p*y^n, where p = positive integer
Thus, x^n = (p*y^n)^n = [p^n]*[y^(n^2)]
We see that [p^n]*[y^(n^2)] is divisible by y. Sufficient.
2) \(x^y\) is divisible by \(y\).
Say x^y = p*y where p = positive integer
Thus, x = [p^1/y]*[y^1/y]
Thus, x^n = [p^1/y]^n*[y^1/y]^n = [p^n/y]*[y^n/y]
[p^n/y]*[y^n/y] is divisible by y if n is a factor of y, else not. No unique answer. Insufficient.
The correct answer: A
Hope this helps!
-Jay
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