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# If a, b, c is an integer, what is value of a+b+c?

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If a, b, c is an integer, what is value of a+b+c?  [#permalink]

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14 Feb 2017, 19:31
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Difficulty:

95% (hard)

Question Stats:

40% (01:26) correct 60% (01:22) wrong based on 179 sessions

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If $$a,b,c$$ is an integer, what is value of $$a+b+c$$?

(1) $$a^2+b^2+c^2=27$$
(2) $$ab+ac+bc=11$$

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Re: If a, b, c is an integer, what is value of a+b+c?  [#permalink]

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14 Feb 2017, 22:16
1
1
(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ac)

St1: a^2 + b^2 + c^2 = 27. Clearly insufficient.

St2: ab + ac + bc = 11. Clearly insufficient.

Combining St1 and St2, (a + b + c)^2 = 49 --> a + b + c = 7 or -7.
Not Sufficient.

CEO
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Re: If a, b, c is an integer, what is value of a+b+c?  [#permalink]

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15 Feb 2017, 10:25
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1
ziyuenlau wrote:
If a,b, and c are integers, what is value of $$a+b+c$$?

(1) $$a^2+b^2+c^2=27$$
(2) $$ab+ac+bc=11$$

Nice question.

Target question: What is the value of a + b + c?

Given: a,b, and c are integers

Statement 1: a² + b² + c² = 27
Since we're told that a,b, and c are INTEGERS, our options for a, b, and c are quite limited.
That said, there is more than one possible solution to the equation in statement 1. Here are two:
Case a: a = 1, b = 1 and c = 5 (notice that a² + b² + c² = 1² + 1² + 5² = 27). In this case, a + b + c = 1 + 1 + 5 = 7
Case b: a = -1, b = -1 and c = -5 (notice that a² + b² + c² = (-1)² + (-1)² + (-5)² = 27). In this case, a + b + c = (-1) + (-1) + (-5)= -7
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: ab + ac + bc = 11
NOTE: Rather than try to find solutions to this equation, I'll start by checking whether the two solutions I found for statement 1 also satisfy statement 2.
Yes, it turns out they DO!
That is, the following solutions satisfy the equation in statement 2:
Case a: a = 1, b = 1 and c = 5 (notice that ab + ac + bc = (1)(1) + (1)(5) + (1)(5) = 11). In this case, a + b + c = 1 + 1 + 5 = 7
Case b: a = -1, b = -1 and c = -5 (notice that ab + ac + bc = (-1)(-1) + (-1)(-5) + (-1)(-5) = 11). In this case, a + b + c = (-1) + (-1) + (-5)= -7
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Since we were able to find the same counter-examples in both statement 1 AND statement 2, the same counter-examples will work when we COMBINE both statements.
That is, the following solutions satisfy BOTH statements:
Case a: a = 1, b = 1 and c = 5. In this case, a + b + c = 1 + 1 + 5 = 7
Case b: a = -1, b = -1 and c = -5. In this case, a + b + c = (-1) + (-1) + (-5)= -7
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Cheers,
Brent
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Re: If a, b, c is an integer, what is value of a+b+c?  [#permalink]

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15 Feb 2017, 11:23
(1)
5^2+1^2+1^2 = 27
3^2+3^2+3^2 = 27
Insuff
(2)
only possible option for positive is (a)5+(b)5+(c)1 = 11
However no information is given if a,b,c are positive integers, hence Ans is E.

Vyshak
I think such kind of problem can be considered as "Hard".
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Re: If a, b, c is an integer, what is value of a+b+c?  [#permalink]

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24 Jul 2018, 06:01
Hi.
About this question, I think it may be possible to solve this further. At least, I gave it a shot and ended up here. Do let me know if this makes sense.

It's ruled out that the two statements by themselves are able to provide solutions so I'm just going to combine the two.

a^2+b^2+c^2 can be written as [(a+b) + c]^2 - (something)

If you expand it and apply the [x+y]^2 formula twice, you get a^2 + b^2 + c^2 +2ab + 2bc +2ac

From the given equations (i) and (ii), we can substitute the values for the above terms as 27 + 22

This gives us (a+b+c)^2 = 27 + 22 = 49.

So (a+b+c) = sqrt(49) = +/- 7

The answer is still E but it saved a valuable amount of time.
Re: If a, b, c is an integer, what is value of a+b+c? &nbs [#permalink] 24 Jul 2018, 06:01
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