If \(x\) and \(y\) are positive integers, is the product \(xy\) divisible by 9?
(1) The product \(xy\) is divisible by 6.so product \(xy\) can have values raging \({6, 12, 18, 24, 30,36.............. }\)
out of all these values only 18, 36,...........are divisible by 9 ........................thereby giving
yes to the above question
and rest are divisible by only 3 not 9...........................................................thereby giving
no to the above question
Since we have values that give both yes and no answers ,
The statement 1 is insufficient.(2) x and y are perfect squares. Let \(x=a^2\) and \(y=b^2\) where a, b are integers.
so \(x\) and \(y\) can be 1, 4, 9, 16.....................
\(xy\) can be 4, 9, 16, 36, 64.............
Even here we have values that give both yes and no answers,
The statement 2 is insufficient.Combining both the statements (1) & (2)The product \(xy\) is divisible by 6
\(x\) and \(y\) are perfect squares.
\(x\) and \(y\) are perfect squares according statement 2 so product \(xy=(ab)^2\)
all the squares in the set {6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, ..............144.............. } are taken into consideration.
i.e., 36, 144...........all of which are divisible by 9.....i.e, definitely YES
Therefore data is sufficient here and Ans is C.