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LCM GCF

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by kop » Wed Nov 20, 2013 12:23 am
The Least Common Multiple of three numbers is four times their Greatest Common Factor. Which of the following must be true of the numbers?

I. At least one of the numbers is odd.
II. Two of the three numbers must be same.
III. At least one number is the same as Greatest Common Factor.
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Source: — Data Sufficiency |
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by theCodeToGMAT » Wed Nov 20, 2013 12:59 am
A = (GCD) * _
B = (GCD) * _
C = (GCD) * _

LCM = 4 * GCD

A = (GCD)
B = (GCD) * (2)
C = (GCD) * (2)^2

Or
A = (GCD)
B = (GCD)
C = (GCD) * (2)^2

Or
A = (GCD)
B = (GCD) * (2)^2
C = (GCD) * (2)^2

I. At least one of the numbers is odd.
--> No

II. Two of the three numbers must be same.
--> "must" - NO

III. At least one number is the same as Greatest Common Factor.
--> YES

So, (III)

Is it the answer?
Last edited by theCodeToGMAT on Wed Nov 20, 2013 9:04 pm, edited 2 times in total.
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by [email protected] » Wed Nov 20, 2013 12:03 pm
Hi kop,

In Roman Numeral questions, it's important to have the answer choices as reference. The way the answers are written will likely allow you to "skip" some of the math (and not have to deal with all 3 Roman Numerals). It's also worth noting that it's easier to prove what's NOT TRUE in these types of situations.

I. At least 1 is odd.

If we use 2, 2, and 8 then the LCM is 4 times the GCF.
Roman Numeral 1 is NOT TRUE

II. Two of the three numbers must be the same.

If we use 1, 2 and 4, then the LCM is 4 times the GCF
Roman Numeral 2 is NOT TRUE

III. At least one of the numbers = the GCF

I can't seem to find a way to eliminate this one, so I assume that it's true.

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by Mathsbuddy » Fri Nov 22, 2013 8:55 am
This is what I reckon:

3 values A, B, C must be composed of factors a, b, c and F:

A = 2 * a * F
B = 2 * b * F
C = 1 * c * F

where F = highest common factor

LCM = 4F, therefore a = b = c = 1

So,

A = 2 * F
B = 2 * F
C = 1 * F

I. At least one of the numbers is odd - Not necessarily as F can be odd or even.
II. Two of the three numbers must be same - True (A and B)
III. At least one number is the same as Greatest Common Factor - True (C)
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by Mathsbuddy » Fri Nov 22, 2013 9:13 am
CORRECTION:

I stand corrected by counter examples!
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by [email protected] » Fri Nov 22, 2013 7:17 pm
Hi Mathsbuddy,

In Quant questions that ask for "what MUST be true", you have to be thorough enough to prove what "COULD be true" vs. what "MUST be true." That difference will be the difference between getting the question right and getting it wrong.

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