Official Solution:What is the value of \(xy\)? Notice that we are not told that \(x\) and \(y\) are integers. (1) \(3^x*5^y=75\).
\(3^x5^y=75\), can be rewritten as \(3^x5^y=3^15^2\). If \(x\) and \(y\) are integers, then \(x=1\) and \(y=2\). However, if they are not integers, an infinite number of solutions would exist, since any value of \(x\) would have a corresponding non-integer value of \(y\) that would satisfy the equation and vice versa. For example, if \(y=1\), then \(3^x5^y=3^x5=75\), hence \(3^x=15\) and \(x\) would be some irrational number, approximately equal to \(2.5\). Therefore, statement (1) alone is not sufficient to determine the value of \(xy\).
(2) \(3^{(x-1)(y-2)}=1\).
This above implies that \((x-1)(y-2)=0\). Thus, either \(x=1\) and \(y\) can be any number (including \(2\)), or \(y=2\) and \(x\) can be any number (including \(1\)). However, this statement alone does not give us a unique solution for \(xy\). Therefore, statement (2) alone is not sufficient.
(1)+(2) If from (2) \(x=1\), then from (1) we have \(3^x5^y=3*5^y=75\), which gives us \(y=2\). If from (2) \(y=2\), then from (1) we have \(3^x5^y=3^x*25=75\), which gives us \(x=1\). Thus, we have a unique solution of \(x=1\) and \(y=2\), and the value of \(xy\) is \(1*2=2\). Sufficient.
Answer: C
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