Is k the square of an integer?

(1) \(k= t^2 * q^6 *r^{10}\), where t, q and r are integers

(2) \(k= t^2 * m^{15}\), where t and m are integers

Is k the square of an integer?

We need to determine whether k is a perfect square.

(1) k = t^2*q^6*r^10, where t, q, and r are integers --> \(k=(tq^3r^5)^2=integer^2\). Sufficient.

(2) k = t^2*m^15, where t and m are integers --> if m is a perfect square itself (say if m=n^2 for some integer n), then k will also be a perfect square but if m is NOT a perfect square, then k also will NOT be a perfect square. For example consider m=1 and m=2. Not sufficient.

Answer: A.
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