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

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If a, b, and c are all integers, is ab + bc + ca + a^2 odd?  [#permalink]

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New post 16 Sep 2012, 00:12
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If a, b, and c are all integers, is ab + bc + ca + a^2 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 + a^2 odd?  [#permalink]

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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 + a^2 odd?  [#permalink]

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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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Re: If a, b, and c are all integers, is ab + bc + ca + a^2 odd?  [#permalink]

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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.
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Re: If a, b, and c are all integers, is ab + bc + ca + a^2 odd?  [#permalink]

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New post 11 Aug 2015, 13:32
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.



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Re: If a, b, and c are all integers, is ab + bc + ca + a^2 odd?  [#permalink]

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New post 14 Jan 2017, 06:50
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Is \(ab+bc+ca+a^2\) odd
= a(b + c) + bc + a^2
= a(b + c + a) + bc

St1: a is odd. --> No info about b and c.
Clearly insufficient.

St2: (b+c) is odd. --> Possible when one value is odd and the other value is even.
Hence if b + c is odd then b *c is even.

Consider a(b + c + a) + bc --> If a is even then the expression is Even + Even = Even.
If a is odd then the expression is --> Odd*(Odd + Odd) + Even = Even + Even = Even.
Thus, \(ab+bc+ca+a^2\) is always even.
Sufficient.

Answer: B
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Re: If a, b, and c are all integers, is ab + bc + ca + a^2 odd?  [#permalink]

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New post 14 Jan 2017, 06:54
roastedchips 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.


STEM

a(b+c+a) + bc = ??

from 1

clearly insuff

from 2

b+c = odd thus one of them is even and the other is odd

now in a(b+c+a) , we know b+c = odd thus if a is odd the b+c+a = even and thus a(b+c+a) is even too and if a is even then a(a+b+c) is even too , i,e, in all cases it is even
bc = even ( since b+c = odd) thus a(b+c+a) +bc is always even and the answer to the question is definite no

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Re: If a, b, and c are all integers, is ab + bc + ca + a^2 odd?  [#permalink]

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New post 15 Jan 2017, 05:21
If a, b, and c are all integers, is ab+bc+ca+(a*a) odd?

(1) a is odd.

Clearly Insufficient. We do not know any info about other terms. It could Even or Odd.

(2) (b+c) is odd.

Re-arrange the question:

ab+ac+bc+a^2 = a (b+c)+ bc+a^2

when (b+c) = odd.......it means one is even and other is odd........So 'bc' is always Even

Let examine the question: a (b+c)+ bc+a^2

Let a =Odd........................O*O+E+O= O+E+0= EVEN

Let a =Even.......................E *O+E+E=E+E+E=EVEN

In both cases the answer to question is NO

Sufficient

Answer: B
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Re: If a, b, and c are all integers, is ab + bc + ca + a^2 odd?  [#permalink]

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New post 05 Apr 2017, 07:21
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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.



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?

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:

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Re: If a, b, and c are all integers, is ab + bc + ca + a^2 odd?  [#permalink]

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New post 21 Dec 2017, 11:11
Hi All,

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

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Re: If a, b, and c are all integers, is ab + bc + ca + a^2 odd? &nbs [#permalink] 21 Dec 2017, 11:11
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