ziyuenlau wrote:If a, b, and c are all integers, is ab + bc + ca + a² odd?
(1) a is odd.
(2) (b+c) is odd.
Target question: Is ab + bc + ca + a² odd?
This is a good candidate for rephrasing the target question.
ab + bc + ca + a² = b(a + c) + a(c + a)
= (b + a)(c + a)
So, we get....
REPHRASED target question: Is (b+a)(c+a) odd?
Aside: Here's a video with tips on rephrasing the target question: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100
When I SCAN the two statements, I see that it might be useful to systematically list the possible outcomes. To do this, I'll list each possible case, and plug in 0 for any EVEN integer and plug in 1 for any ODD integer. We get:
case a: a = odd, b = odd, c = odd. Here, (b+a)(c+a) = EVEN
case b: a = odd, b = odd, c = even. Here, (b+a)(c+a) = EVEN
case c: a = odd, b = even, c = odd. Here, (b+a)(c+a) = EVEN
case d: a = odd, b = even, c = even. Here, (b+a)(c+a) = ODD
case e: a = even, b = odd, c = odd. Here, (b+a)(c+a) = ODD
case f: a = even, b = even, c = odd. Here, (b+a)(c+a) = EVEN
case g: a = even, b = odd, c = even. Here, (b+a)(c+a) = EVEN
case h: a = even, b = even, c = even. Here, (b+a)(c+a) = EVEN
Statement 1: a is odd
This means we're dealing with case a, b, c, or d
For cases a, b and c,
(b+a)(c+a) is EVEN
For case d,
(b+a)(c+a) is ODD
Since we cannot answer the
REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: (b+c) is odd
This means we're dealing with case b, c, f or g
In ALL of these cases,
(b+a)(c+a) is EVEN
This means we can be certain that
(b+a)(c+a) is EVEN
Since we can answer the
REPHRASED target question with certainty, statement 2 is SUFFICIENT
Answer:
B
Cheers,
Brent
