Lucky2783 wrote:
x and y are positive integers such that x < y. If \(x\sqrt{y} = 6\sqrt{6}\), then xy could equal
A. 36
B. 48
C. 54
D. 96
E. 108
Square both sides:
\(x^2y = 216\)
Let's just try a few values.
x = 1, y = 216, xy = 216 ... not an answer choice.
x = 2, y = 216/4 = 54, xy = 108 ... Yay!
That is ALL we need to do. No reason to get all fancy.
Answer choice E.
.
.
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Bonus:
x = 3, y = 216/9 = 24, xy = 72 ... not an answer choice.
x = 4, y = 216/16 = 13.something ... invalid since y is not an integer.
x = 5, y = 216/25 = 8.something ... invalid since y is not an integer.
x = 6, y = 216/36 = 6 ... invalid since x is not less than y.
If we keep going x will be greater than y. We didn't need to test all of these, but even if we'd had to, it wasn't that bad. Brute force and trial-and-error are frequently (usually) faster than figuring out formulas.
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