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# a, b, and c are integers such that ab + c = 7, ac + b = 5, a

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Joined: 07 Jun 2014
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a, b, and c are integers such that ab + c = 7, ac + b = 5, a [#permalink]  18 Mar 2018, 06:17
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Question Stats:

40% (02:33) correct 60% (03:21) wrong based on 5 sessions
a, b, and c are integers such that ab + c = 7, ac + b = 5, and a + b + c = 6. What is the value of abc ?

A. 2
B. 6
C. 9
D. 12
E. 18

Drill 2
Question: 10
Page: 499
[Reveal] Spoiler: OA

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Sandy
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GRE Prep Club Legend
Joined: 07 Jun 2014
Posts: 4856
GRE 1: Q167 V156
WE: Business Development (Energy and Utilities)
Followers: 102

Kudos [?]: 1731 [0], given: 397

Re: a, b, and c are integers such that ab + c = 7, ac + b = 5, a [#permalink]  18 Mar 2018, 07:31
Expert's post
Explanation

By rearranging the first two equations, you obtain c = 7 – ab and b = 5 − ac.

By substituting these relationships into the third equation, a + (5 − ac) + (7 − ab) = 6 is obtained.

Manipulating the equation yields a − ac − ab + 12 = 6, or a − a (c + b) + 6 = 0. Since a + b + c = 6, b + c = 6 − a.

Substituting this into the previous equation yields $$a - a (6 - a) + 6 = a - 6a + a^2 + 6 = 0$$, or $$a^2 - 5a + 6 = 0$$.

Factoring and finding the roots gives you a = 2 or a = 3.

If a = 2—by substituting the value back into the equations—then b = 2 and c = 1. If a = 3, then b = 3 and c = 1. You are looking for abc, and either way, the value is 3 × 2 × 1 = 6. The answer is choice (B).
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Sandy
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Manager
Joined: 15 Feb 2018
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Re: a, b, and c are integers such that ab + c = 7, ac + b = 5, a [#permalink]  22 Mar 2018, 00:24
Solving this question is very time-consuming.
GRE Prep Club Legend
Joined: 07 Jun 2014
Posts: 4856
GRE 1: Q167 V156
WE: Business Development (Energy and Utilities)
Followers: 102

Kudos [?]: 1731 [0], given: 397

Re: a, b, and c are integers such that ab + c = 7, ac + b = 5, a [#permalink]  22 Mar 2018, 02:02
Expert's post
gremather wrote:
Solving this question is very time-consuming.

This can be solved by plugging, but it is quite risky and needs a good grasp of number theory in general.

You might be able to realize by observing that numbers a, b and c are either positive negative integers less than 6. So picking numbers like 1, 2, 3 etc can help.
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Re: a, b, and c are integers such that ab + c = 7, ac + b = 5, a   [#permalink] 22 Mar 2018, 02:02
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