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If a, b, and c are all integers, is ab+bc+ca

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If a, b, and c are all integers, is ab+bc+ca [#permalink] New post 16 Sep 2012, 00:12
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If a, b, and c are all integers, is ab+bc+ca+(a*a) odd?

(1) a is odd.
(2) (b+c) is odd.



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Re: If a, b, and c are all integers, is ab+bc+ca [#permalink] New post 16 Sep 2012, 01:01
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navigator123 wrote:
If a, b, and c are all integers, is ab+bc+ca+(a*a) odd?

(1) a is odd.
(2) (b+c) is odd.
- From HULT free tests


From Question stem : a^2+ab+ac+bc
=(a+c)(a+b). product of 2 numbers.
St1: Insufficient: Let say a is odd, c even & b even, so (a+b) =odd & (a+b)=odd , So (a+b)(a+c) = odd, which gives YES
Let say a is odd, c odd & b even, So (a+b)= odd & (a+c)=even, So (a+b)(a+c)= even, which gives NO.

St 2: Sufficient: if b+c = odd, any one of b & c should be even and another odd. so a can be either odd or even. Let say b=O & C=E and A=O
(a+b)=E, (a+c) = E, hence (a+b)(a+c)= E, which gives NO
Now Let say b=O & C=E and A=E, (a+b)=O, (a+c)=E, Hence (a+b)(a+c)= E, which gives NO
So any one of the factors (a+b) or (a+c) will be either E or O which will always give an Even number.
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If a, b, and c are all integers, is ab+bc+ca+a2 odd? [#permalink] New post 19 Feb 2014, 16:05
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If a, b, and c are all integers, is ab+bc+ca+a2 odd?

(1) a is odd.
(2) (b+c) is odd.
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Re: If a, b, and c are all integers, is ab+bc+ca+a2 odd? [#permalink] New post 19 Feb 2014, 21:59
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mhknair wrote:
If a, b, and c are all integers, is ab+bc+ca+a2 odd?

(1) a is odd.
(2) (b+c) is odd.


Looking at the statements, I would try to club b and c together.

Given Expression: a(b+c) + a^2 + bc
There are 3 terms: a(b+c), a^2, bc

(1) a is odd.
a^2 is certainly odd. But we don't know anything about the other two terms.
Say b and c are both even. Then 2 terms (a(b+c) and bc) are even and one (a^2) is odd so sum is odd.
Say b and c are both odd. Then 2 terms (a^2 and bc) are odd and one (a(b+c)) is even so sum is even.
Not sufficient.

(2) (b+c) is odd.
If b+c is odd, it means one of b and c is odd and the other is even. So bc will be even
Now if a is odd, two terms are odd (a(b+c) and a^2) while the third term (bc) is even. So sum will be even.
If a is even, all three terms are even so sum will be even.
In any case, the sum will be even so this statement alone is sufficient.

Answer (B)
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Re: If a, b, and c are all integers, is ab+bc+ca+a2 odd? [#permalink] New post 19 Feb 2014, 23:49
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Re: If a, b, and c are all integers, is ab+bc+ca+a2 odd? [#permalink] New post 20 Feb 2014, 00:18
mhknair wrote:
If a, b, and c are all integers, is ab+bc+ca+a2 odd?

(1) a is odd.
(2) (b+c) is odd.


Simpler: Regroup to get (ab + ac) + bc + a^2. Since a is odd, we know \underbrace{ab + ac}_{odd} + bc + \underbrace{a^2}_{odd}. Next, since b+c is odd we know either b or c is odd and the other is even -- so bc must be even. Odd + even + odd is even.

Alternatively: Regroup to get (a^2 + ab + ac) + bc. Since a and (b+c) are odd, we know a+b+c is even and \underbrace{a(a+b+c)}_{even} + bc. Next, since b+c is odd we know b or c is odd and the other is even -- so bc must be even. Even + even is even.
Re: If a, b, and c are all integers, is ab+bc+ca+a2 odd?   [#permalink] 20 Feb 2014, 00:18
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