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If a and b are positive even integers, and the least common

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vikramm
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If a and b are positive even integers, and the least common [#permalink] New post   Updated on: 28 Aug 2013, 04:21
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If a and b are positive even integers, and the least common multiple of a and b is expressed as a*b/n, which of the following statements could be false?

A. n is a factor of both a and b
B. (a*b)/n < ab
C. ab is multiple of 2.
D. (a*b)/n is a multiple of 2.
E. n is a multiple of 4.
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E

Originally posted by vikramm on 11 Oct 2005, 18:32.
Last edited by Bunuel on 28 Aug 2013, 04:21, edited 1 time in total.
Renamed the topic and edited the question.
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blueseas
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Re: This one from Kaplan.... If 'a' and 'b' are positive even [#permalink] New post   28 Aug 2013, 04:12
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vikramm wrote:
This one from Kaplan....

If 'a' and 'b' are positive even integers, and the least common multiple of 'a' and 'b' is expressed as ab/n, which of the following could be false?
A) n is a factor of both a and b
B) ab/n < ab
C) ab is a multiple of 2
D) ab/n is a multiple of 2
E) n is a multiple of 4


This question is easier if you know that, for any positive integers a and b:

LCM(a,b) * GCD(a,b) = a*b

So LCM(a,b) = ab/GCD(a,b)

In this question, n is just the GCD of a and b. So we know automatically that n is a factor of a and b... A) must be true.
If a and b are both even, they are both divisible by 2, so the smallest possible value for their GCD is 2. That is, n >= 2, so B) must be true
If a and b are even, ab is clearly even, and so is the LCM of a and b, so C) and D) must be true.
E) is the only one that might be false; E will only be true if both a and b are divisible by 4.

hence E
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Re: If a and b are positive even integers, and the least common [#permalink] New post   23 Jan 2015, 13:56
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Hi All,

The last poster in this thread (kinjiGC) had some concerns about whether this question was "valid" or not, but I can assure you that it is.

The question is based on Number Property rules that exist in the realm of factors and multiples. Given the restrictions in the prompt, 4 of the answers are ALWAYS TRUE....

For example, since we're told that A and B are POSITIVE EVEN INTEGERS, we know that (A)(B) = (even)(even), so the product will ALWAYS be EVEN. Thus AB is a multiple of 2 and we can eliminate Answer C (since we're asked to find something that is SOMETIMES FALSE - meaning NOT ALWAYS TRUE).

These types of "...must be true..." questions almost always involve Number Properties of some kind, so you can TEST VALUES or use Number Property Rules to get to the correct answer. You'll see a few questions in this general "format" on Test Day and a bunch of questions involving Number Properties in general (especially in DS).

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Re: This one from Kaplan.... If 'a' and 'b' are positive even [#permalink] New post   19 Jan 2014, 07:40
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blueseas wrote:
vikramm wrote:
This one from Kaplan....

If 'a' and 'b' are positive even integers, and the least common multiple of 'a' and 'b' is expressed as ab/n, which of the following could be false?
A) n is a factor of both a and b
B) ab/n < ab
C) ab is a multiple of 2
D) ab/n is a multiple of 2
E) n is a multiple of 4



E) is the only one that might be false; E will only be true if both a and b are divisible by 4.

hence E


Any information is provided such that a and b cannot be a multiple of 4 ( 'a' and 'b' are positive even integers - a can be 4 and b can be 16 or 8) ? I don't think it is valid question.
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Re: If a and b are positive even integers, and the least common [#permalink] New post   24 Jan 2015, 13:01
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If a and b are positive even integers, and the least common multiple of a and b is expressed as a*b/n, which of the following statements could be false?

A. n is a factor of both a and b
B. (a*b)/n < ab
C. ab is multiple of 2.
D. (a*b)/n is a multiple of 2.
E. n is a multiple of 4.

If we divide by n and get a multiple of a and of b, then n must be a common factor. Answer A must be true.

If a and b are even integers, dividing by a factor (also an integer) results in something smaller than the product ab. Answer B must be true.

If either a or b is even, then ab must be even. Answer C must be true.

If both are even, the smallest either number can be is 2. Thus, the smallest common factor must be 2. Answer D must be true.

If b is 6 and a is 2, n would be 6, which is not 4. E can be false.
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Re: If a and b are positive even integers, and the least common [#permalink] New post   15 Jun 2016, 00:03
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Using a=2, b=4. LCM (2,4) = 4. This means ab/n can be written as 2*4/2 (because LCM(=4) the least common multiple of 'a' and 'b' is expressed as ab/n, 2*4/2=4=LCM). Thus n=2 (and a=2, b=4)
A. n is a factor of both a and b - True. 2 is factor for 2 and 4
B. (a*b)/n < ab - True (4<8)
C. ab is multiple of 2. - True (8 is multiple of 2)
D. (a*b)/n is a multiple of 2. - True (4 is multiple of 2)
E. n is a multiple of 4. False (n=2 is not multiple of 4)
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Re: If a and b are positive even integers, and the least common [#permalink] New post   31 Jul 2016, 05:22
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manuchadha wrote:
Using a=2, b=4. LCM (2,4) = 4. This means ab/n can be written as 2*4/2 (because LCM(=4) the least common multiple of 'a' and 'b' is expressed as ab/n, 2*4/2=4=LCM). Thus n=2 (and a=2, b=4)
A. n is a factor of both a and b - True. 2 is factor for 2 and 4
B. (a*b)/n < ab - True (4<8)
C. ab is multiple of 2. - True (8 is multiple of 2)
D. (a*b)/n is a multiple of 2. - True (4 is multiple of 2)
E. n is a multiple of 4. False (n=2 is not multiple of 4)


Great Explanation ! Thanks :) !
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[#permalink] New post   11 Oct 2005, 19:08
E may be false.
If a=4, b=2 least comm. multiple is 4.
ab/n=4 => n=2 which is not a multiple of 4
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Re: If a and b are positive even integers, and the least common [#permalink] New post   22 Nov 2023, 08:20
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Re: If a and b are positive even integers, and the least common [#permalink]
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