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 Post subject: GMAT Number TheoryPosted: Thu Feb 03, 2011 8:54 pm

Joined: Sun May 30, 2010 3:15 am
Posts: 424
The least common multiple of positive integer m and 3-digit integer n is 690. If n is not divisible by 3 and m is not divisible by 2, what is the value of n?

A. 115
B. 230
C. 460
D. 575
E. 690

(B) We factorize 690 = 2 × 3 × 115 = 2 × 3 × 5 × 23.
Dividing 690 starting by the smallest factor we get possible values for 3-digit integer n:
690, 690 / 2 = 345, 690 / 3 = 230, 690 / 5 = 138, 690 / 6 = 115.

690 / 10 = 69, which is a 2-digit number, therefore n must be one of the following:
690, 345, 230, 138, 115.

n is not divisible by 3. So we eliminate 690, 345, 138.

m is not divisible by 2, but the least common multiple of m and n is. So n must be divisible by 2. We eliminate 115.

That leaves us only one option for n, 230. The correct answer is B.
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Is the reason that C and D are not correct because C implies 460 × 1.5 and D implies 575 × 1.2? Neither 1.5 or 1.2 are integers?

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 Post subject: Re: math (test 4, question 11): number theory, LCMPosted: Thu Feb 03, 2011 9:08 pm

Joined: Sun May 30, 2010 2:23 am
Posts: 498
Yes, you're right.
The answer choices C and D are NOT the factors of 690. Therefore we can eliminate them, since the LCM of any numbers must be divisible by those numbers.

Furthermore,
Let's try to solve this question by considering the answer choices.
We can eliminate choices C and D as described above.
The answer choice E is divisible by 3, while the original statement tells us that n is NOT divisible by 3. So we can eliminate this answer choice as well.

It's a little bit harder to consider the answer choices A and B. The reasoning we use is the same as in the original explanation.

Let's consider A. Suppose n is 115. Then m must be divisible by 690/115 = 6. Therefore m must be divisible by 2 (since 6 = 2 × 3). But the original statement tells us that m is NOT divisible by 2. So we eliminate the answer choice A.

B is the only answer choice that's left. So B is the correct answer.

P.S. If n is 230, then m, which is odd, can be one of the following:
3, 15, 69, 345.

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