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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
Joined: 16 Aug 2015
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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, 01:01
00:00

Difficulty:

65% (hard)

Question Stats:

47% (01:20) correct 53% (01:05) wrong based on 75 sessions

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

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MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Senior CR Moderator Status: Long way to go! Joined: 10 Oct 2016 Posts: 1390 Location: Viet Nam Re: If a, b, and c are different positive integers, what is the value of a [#permalink] ### Show Tags 08 Sep 2017, 02:37 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. Answer D. _________________ Senior Manager Joined: 29 Jun 2017 Posts: 499 GPA: 4 WE: Engineering (Transportation) If a, b, and c are different positive integers, what is the value of a [#permalink] ### Show Tags 08 Sep 2017, 02:44 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 DEFINITE ANSWER 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 HENCE D is the ANSWER _________________ Give Kudos for correct answer and/or if you like the solution. Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 5600 GMAT 1: 800 Q59 V59 GPA: 3.82 Re: If a, b, and c are different positive integers, what is the value of a [#permalink] ### Show Tags 10 Sep 2017, 18:22 => 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. Therefore the answer is D Ans: D _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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Manager
Joined: 30 Mar 2017
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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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19 Apr 2018, 16:32
I think plugging numbers is the best approach here.

a,b,c are different positive integers, so let's pick the smallest values for each --> a=1, b=2, c=3

Statement 1
$$a^2+b^2+c^2=1^2+2^2+3^2=1+4+9=14$$
If a,b,c are anything other than 1,2,3, then $$a^2+b^2+c^2>14$$
Thus, a+b+c must equal 6.

Statement 2
$$ab+bc+ca=1*2+2*3+3*1=2+6+3=11$$
If a,b,c are anything other than 1,2,3, then $$ab+bc+ca>11$$
Thus, a+b+c must equal 6.

Re: If a, b, and c are different positive integers, what is the value of a   [#permalink] 19 Apr 2018, 16:32
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