GMAT Problem 1

p, q, and r are integers and p + q + r is an odd number. Which of the following must always be true?

(i) p² + q² + r² is odd
(ii) p - q + r is odd
(iii) pq + qr + rp is odd

(A) i and ii
(B) i and iii
(C) ii and iii
(D) i only
(E) iii only

Let's
evaluate either (i) or (iii) first since these two appear most
frequently. Since the expression in (i) is simpler, it should be first.

If p + q + r
is odd, then it is either the sum of three odd numbers, or one odd and
two even numbers. When you square a number, it stays either odd or
ever, so the expression p² + q² + r² is also the sum of either three odd numbers, or one odd and two even numbers. Thus p² + q² + r² is odd. This means we can cross out choices C and E.

The next statement to evaluate is (ii), since it is the simplest. Notice that p - q + r = (p + q + r) - 2q. The expression (p + q + r) - 2q is an odd number minus and even number, hence an odd number. Therefore (ii) is true as well.

So
the correct answer is A, and we didn't even have to check (iii). As an
exercise for extra practice, you should verify that if p + q + r is odd, then pq + qr + rp can be even sometimes.