sanju09 wrote:Is (a^b) (c^d) odd, where a, b, c, and d are positive integers?
I. a is even and b is odd.
II. c is odd and d is even.
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(1) a is even and b is odd.
If a = 2, b = 1, then (a^b) * (c^d) = 2 * (c^d), which implies that a^b will always be even.
It is given that c and d are positive integers, so c^d will either be an odd or an even integer.
If c^d = even, then (a^b) * (c^d) = even * even = even
If c^d = odd, then (a^b) * (c^d) = even * odd = even
Therefore, (a^b) * (c^d) is always an even integer. So, the answer to the main question is "no"; SUFFICIENT.

(2) c is odd and d is even.
If c = 1 and d = 2, then c^d = 1, odd
So, c^d will always be odd.
It is given that a and b are positive integers, so a^b will either be an odd or an even integer.
If c^d = even, then (a^b) * (c^d) = even * odd = even
If c^d = odd, then (a^b) * (c^d) = odd * odd = odd
We do not get a definite answer; NOT sufficient.

The correct answer is A.