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a, b, and c are positive integers with abc + 2ab + 2bc + 2ca + 4a + 4b + 4c = 447. What is the value of a + b + c?

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a, b, and c are positive integers with abc + 2ab + 2bc + 2ca + 4a + 4b + 4c = 447. What is the value of a + b + c?

by Max@Math Revolution » Fri Apr 17, 2020 6:35 pm

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[GMAT math practice question]

a, b, and c are positive integers with abc + 2ab + 2bc + 2ca + 4a + 4b + 4c = 447. What is the value of a + b + c?

A. 17
B. 19
C. 21
D. 23
E. 25
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Re: a, b, and c are positive integers with abc + 2ab + 2bc + 2ca + 4a + 4b + 4c = 447. What is the value of a + b + c?

by Max@Math Revolution » Sun Apr 19, 2020 7:32 pm
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abc + 2ab + 2bc + 2ca + 4a + 4b + 4c + 8 = 455
a(bc + 2b + 2c + 4) + 2(bc + 2b + 2c + 4) = 455 (taking out a common fraction of a from the first 4 terms and a common factor of 2 from the last 4 terms)
= (a + 2)(bc + 2b + 2c + 4) = 455 (taking out a common factor of (bc + 2b + 2c +4))
(a + 2)[b(c + 2) + 2(c + 2)] = 455 (taking out common factors of b and 2)
= (a + 2)(b + 2)(c + 2) = 455 (taking out a common factor of (c + 2))
= 3*5*11.
a, b, and c are interchangeable since the equation (a + 2)(b + 2)(c + 2) = 5*7*13 is symmetric.
Then we have a + b + c = 3 + 5 + 11 = 19.

Therefore, B is the answer.
Answer: B
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