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# If a, b, and c are different positive integers, what is the value of a

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Math Revolution GMAT Instructor
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If a, b, and c are different positive integers, what is the value of a [#permalink]

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08 Sep 2017, 00:01
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[GMAT math practice question]

If a, b, and c are different positive integers, what is the value of a+b+c?

1) $$a^2+b^2+c^2=14$$
2) $$ab+bc+ca=11$$
[Reveal] Spoiler: OA

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Re: If a, b, and c are different positive integers, what is the value of a [#permalink]

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08 Sep 2017, 01:37
MathRevolution wrote:
[GMAT math practice question]

If a, b, and c are different positive integers, what is the value of a+b+c?

1) $$a^2+b^2+c^2=14$$
2) $$ab+bc+ca=11$$

(1) $$a^2+b^2+c^2=14$$

Since $$a,b,c$$ are different possitive integers, we need to divide 14 into three perfect square.

Note that $$4^2=16 > 14$$, so we only have 3 possible values: 1, 2 and 3.

$$1^2+2^2+3^2=14$$. Perfect.

$$a+b+c=1+2+3=6$$. Sufficient.

(2) $$ab+bc+ca=11$$

Since $$a,b,c$$ are different possitive integers, we have $$abc \geq 1*2*3=6$$

$$11=ab+bc+ca \geq 3\sqrt[3]{a^2b^2c^2} \implies abc \leq \frac{11\sqrt{11}}{3\sqrt{3}} \implies abc \leq 7$$

If $$abc=6$$, we have $$1*2+2*3+3*1=11$$. Perfect.

If $$abc=7$$, since 7 is a prime, we can't find any three different positive numbers.

Hence $$a+b+c=6$$. Sufficient.

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If a, b, and c are different positive integers, what is the value of a [#permalink]

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08 Sep 2017, 01:44
Ans is D :

1) a^2 + b^2 + c^2 = 14
if a,b or c anyone is greater than 3 then it will become >= 16 square of single number
so a,b,c are less than 4 and also positive => greater than zero
also integers => 1,2,3 where a,b,and c can be either of 1,2,3 but no repeats, => SUM = 6
2) ab+bc+ca =14
a,b,c>0
lets us say a =4 and b=1 and c=2 then equation is => 4+2+8 => 16
a=4 b=1 and c=3 => equation is 4+3+12=> 17

means all must be less than 4 again

so again same combination 1,2,3 for a,b,c => sum =6

DEFINITE

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Re: If a, b, and c are different positive integers, what is the value of a [#permalink]

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10 Sep 2017, 17:22
=>
Condition 1)
We can assume a < b < c without loss of generality.
The maximum value of c is 3 and c^2 = 9
a^2 + b^2 = 5.
Then we have b = 2 and a = 1.
a + b + c = 1 + 2 + 3 = 6

Condition 2)
We can assume a < b < c without loss of generality.
ab + bc + ca = (a+b)c + ab = 11
Since a + b >= 3, the maximum value of c = 3.
If c = 3, ab + 3b + 3a = 11 or ab + 3a + 3b + 9 = 20.
We have (a+3)(b+3) = 20.
Then a = 1 and b = 2.
Thus a + b + c = 1 + 2 + 3 = 6.

Ans: D
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Re: If a, b, and c are different positive integers, what is the value of a   [#permalink] 10 Sep 2017, 17:22
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