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GMAT Number Theory (data sufficiency)
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Author:  clover [ Fri Oct 19, 2012 5:18 am ]
Post subject:  GMAT Number Theory (data sufficiency)

After the first term, each term in a sequence is five times greater than half the preceding term. If x is the first term of the sequence, and x does not equal zero, is the value of the fourth term minus the second term an integer?
(1) x is a multiple of 12.
(2) x is a multiple of 56.

A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
D. Either statement BY ITSELF is sufficient to answer the question.
E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(B) Each term is 5/2 multiplied by the preceding term. If x is the first term, then the second term will be 5x/2, the third term will be 25x/4 and the fourth term will be 125x/8.

The fourth term minus the second term = (125x/8) – (5x/2) = (125x/8) – (20x/8) = 105x/8

Statement (1) tells us x is a multiple of 12. For 105x/8 to be an integer, 105x needs to be a multiple of 8. This will only happen if 105x has a factor of 8, or 2 × 2 × 2. Since 105 has no factors of 2, and 12 = 2 × 2 × 3 has two factors of 2, 105x will only be divisible by 8 if x is also a multiple of 8.

Some multiples of 12 are multiples of 8 (like 24) and some aren't (like 12). So statement (1) is insufficient.

Statement (2) tells us that x is a multiple of 56, so it's also a multiple of 8. So 105x/8 will be an integer. The answer is (B).
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For (2) x could be 56 or 56 *2 , and we still don't know the value of the difference. Answer should be E.

Author:  Gennadiy [ Fri Oct 19, 2012 5:22 am ]
Post subject:  Re: GMAT Number Theory (data sufficiency)

Quote:
For (2) x could be 56 or 56 *2 , and we still don't know the value of the difference. Answer should be E.
We are not asked to find the actual value, we are asked if such value is an integer or not.

For any x = 56k (where k is an integer) the value is
105x/8 = 105 × 56k / 8 = 735k
The result, 735k, will be different for different values of k. But it will ALWAYS be an integer.

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