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If a, b, and c are all integers, is ab+bc+ca+ aa^2 odd?

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Source: — Data Sufficiency |

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by Jay@ManhattanReview » Wed Apr 05, 2017 5:38 am
ziyuenlau wrote:If a, b, and c are all integers, is ab+bc+ca+a^2 odd?

(1) a is odd.

(2) (b+c) is odd.

OA = B
Hi ziyuenlau,

Question: Is ab+bc+ca+a^2 odd?

S1: a is odd.

We do not have any information about b and c. We can only be sure that a^2 = odd*odd =odd. Since ab, bc, and ca can be even or odd, it is not possible to deduce whether ab+bc+ca+a^2 is odd.

Let's see it in detail.

ab+bc+ca+a^2 = Odd/Even + Odd/Even +Odd/Even + Odd = Odd/Even. Not sufficient.

S2: (b+c) is odd.

(b+c) is odd implies that one between b and c is odd and the other is even. Say b is odd and c is even.

ab+bc+ca+a^2 = a*Odd + Odd*Even + Even*a + a^2 = a*Odd + Even + Even + a^2

Case 1: If a is odd

a*Odd + Even + Even + a^2 = Odd*Odd + Even + Even + (Odd)^2 = Odd +Even + Even + Odd = Even

Case 1: If a is even

a*Odd + Even + Even + a^2 = Even*Odd + Even + Even + (Even)^2 = Even +Even + Even + Even = Even

Sufficient.

Even if we assume that Say c is odd and b is even, we would reach the same conclusion.

The correct answer: B

Hope this helps!

Relevant book: Manhattan Review GMAT Data Sufficiency Guide

-Jay
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by Brent@GMATPrepNow » Wed Apr 05, 2017 7:14 am
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
Brent Hanneson - Creator of GMATPrepNow.com
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by [email protected] » Wed Apr 05, 2017 3:03 pm
Hi ziyuenlau,

This question can be solved by TESTing VALUES. By extension, if you recognize the Number Properties involved, then you'll only have to TEST 'odds' vs. 'evens' (and NOT have to work through multiple iterations of each option).

We're told that A, B and C are all INTEGERS. We're asked if (A)(B) + (B)(C) + (C)(A) + A^2 is ODD. This is a YES/NO question.

1) A is ODD

With this Fact, we have to consider whether B and C are odds, evens or a mix of the two...

IF... A = 1, then we have.... (1)(B) + (B)(C) + (C)(1) + (1)^2

IF... B=0 and C=0.... (1)(0) + (0)(0) + (0)(1) + (1)^2 = 1 and the answer to the question is YES.
IF... B=1 and C=1.... (1)(1) + (1)(1) + (1)(1) + (1)^2 = 4 and the answer to the question is NO.
Fact 1 is INSUFFICIENT

2) (B+C) is ODD

With this Fact, we know that ONE of these two variables is ODD and the other is EVEN. The "A" can be odd or even...

IF... B=0 and C=1, then we have.... (A)(0) + (0)(1) + (1)(A) + (A)^2

IF.... A=0.... (0)(0) + (0)(1) + (1)(0) + (0)^2 = 0 and the answer to the question is NO.
IF.... A=1.... (1)(0) + (0)(1) + (1)(1) + (1)^2 = 2 and the answer to the question is NO.

IF... B=1 and C=0, then we have.... (A)(1) + (1)(0) + (0)(A) + (A)^2

IF.... A=0.... (0)(1) + (1)(0) + (0)(1) + (0)^2 = 0 and the answer to the question is NO.
IF.... A=1.... (1)(1) + (1)(0) + (0)(1) + (1)^2 = 2 and the answer to the question is NO.
The answer to the question is ALWAYS NO.
Fact 2 is SUFFICIENT

Final Answer: B

GMAT assassins aren't born, they're made,
Rich
Contact Rich at [email protected]
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