Odd & Even

[leumas]

If p, q, and r are integers, is pq + r even?
(1) p + r is even.
(2) q + r is odd.

OA says Ebut I feel correct answer is C

Please clarify.

[GmatMathPro]

Statement 1: If p+r is even, either 1)p=even and r=even or 2)p=odd and r=odd. Since you were down to C or E, I'm going to assume you see why this is insufficient.

Statement 2: If q+r is odd, either 3)q=even and r=odd or 4) q=odd and r=even. Once again, I assume you see why this is insufficient.

Statements 1&2: If both are true at the same time, it would either have to be condition 1 from statement 1 matched with condition 4 from statement 2, in which case we have p=even, q=odd, r=even, pq=even so pq+r=even. OR it could be condition 2 matched with condition 3 in which case p=odd, q=even, r=odd, pq=even and pq+r=odd. Two different answers, so the answer is E

[GMATGuruNY]

If p, q, and r are integers, is pq + r even?
(1) p + r is even.
(2) q + r is odd.

OA says Ebut I feel correct answer is C

Please clarify.

This is an EVEN VS. ODD question.
Try to plug in combinations of values that satisfy both statements.

Let r=1, p=1, q=2:
Statement 1: p+r = 1+1 = 2, which is even.
Statement 2: q+r = 2+1 = 3, which is odd.
Is pq + r even?
pq + r = 1*2 + 1 = 3.
NO.

Let r=2, p=2, q=1:
Statement 1: p+r = 2+2 = 4, which is even.
Statement 2: q+r = 1+2 = 3, which is odd.
Is pq + r even?
pq + r = 2*1 + 2 = 4.
YES.

Since each combination above satisfies both statements, and in the first case the answer is NO and in the second case the answer is YES, insufficient.

The correct answer is E.