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p and q are positive integers and relative primes. Is p divi

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p and q are positive integers and relative primes. Is p divi

by Max@Math Revolution » Sun Nov 10, 2019 11:04 pm
[GMAT math practice question]

p and q are positive integers and relative primes. Is p divisible by 1979?

1) p is a multiple of 1979.
2) p/q = 1 - (1/2) + (1/3) - (1/4) +...- (1/1318) + (1/1319).
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Source: — Data Sufficiency |
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by Max@Math Revolution » Tue Nov 12, 2019 11:15 pm
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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

The question "is p divisible by 1979" is equivalent to condition 1) "p is a multiple of 1979".

Condition 2)
Remember that -1/2k = 1/2k - 1/k for k = 1, 2, 3, ..., 659.
-1/2 = 1/2 - 1/1
-1/4 = 1/4 - 1/2
-1/6 = 1/6 - 1/3
...
-1/1318 = 1/1318 - 1/659
p/q = 1 + 1/2 + 1/3 + 1/4 + ... + 1/1317 + 1/1318 + 1/1319 - 2(1/2 + 1/4 + ... + 1/1318)
= 1 + 1/2 + 1/3 + 1/4 + ... + 1/1317 + 1/1318 + 1/1319 - (1/1 + 1/2 + ... + 1/659)
= 1/660 + 1/661 + ... + 1/1318 + 1/1319
= (1/660 + 1/1319) + (1/661 + 1/1318) + ... + (1/989 + 1/990)
= 1979/(660*1319) + 1979/(661*1318) + ... + 1979/(989*990)
= (1979*k)/(660*661*...*1318*1319)

Then, we have p(660*661*...*1318*1319) = q(1979*k).
Since 1979 is a prime number, p is a multiple of 1979.

Therefore, D is the answer.
Answer: D

This question is a CMT4 (B) question: condition 1) is easy to work with, and condition 2) is difficult to work with. For CMT4 (B) questions, D is most likely to be the answer.
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