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If x and y are positive integers, what is the value of xy?

1) The greatest common factor of x and y is 10 2) the least common multiple of x and y is 180

THanks

Welcome to the Gmat Club. Below is a solution for your problem:

If x and y are positive integers, what is the value of xy?

(1) The greatest common factor of x and y is 10. Clearly insufficient as multiple values are possible for \(xy\): for instance if \(x=y=10\), \(GCF(x,y)=10\) and \(xy=100\) BUT if \(x=10\) and \(y=20\), \(GCF(x,y)=10\) and \(xy=200\).

(2) the least common multiple of x and y is 180. Also insufficient as again multiple values are possible for \(xy\): for instance if \(x=10\) and \(y=180\), \(LCM(x,y)=180\) and \(xy=1800\) BUT if \(x=1\) and \(y=180\), \(LCM(x,y)=180\) and \(xy=180\).

(1)+(2) The most important property of LCM and GCF is: for any positive integers \(x\) and \(y\), \(xy=GCF(x,y)*LCM(x,y)\), hence \(xy=GCF(x,y)*LCM(x,y)=10*180=1800\). Sufficient.

Some solutions above rely on the "property" that GCFxLCM is xy, which is nice when you know the property. Unfortunately, it is nearly impossible to learn all such properties for the GMAT. Here's a away to (try) to derive the property for this and similar problems:

GCF = 10 = 2x5 LCM = 180 = 3x3x2x2x5

The solution centers on the definition of LCM. Remember how we find LCM for two numbers? We take all the prime factors the two numbers share and multiply them by prime factors that the numbers don't share.

ie, LCM of (2x2x5x7) and (2x7x11) would be: (2x2x5x7x11)

Since we are looking for the actual product of x and y, the result will be the LCM times the factors they share (since we didn't double count them in the original LCM calculation), namely the GCF, since that's what the GCF encapsulates.

Re: If x and y are positive integers, what is the value of xy? [#permalink]

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19 Dec 2012, 16:32

1

This post received KUDOS

Expert's post

This is tough.

whenever you see such question break down into factors the numbers.

So we want information to exactly calculate \(X*Y\)

1) we have 10 and the factor are 2 and 5 a bunch of numbers could have 2 and 5 in the shaded region to calculate the GCF (remembre that to obtain the GCF you take between two numbers those have the least power). Insuff

2) the same as above 180 equal \(2^2\) \(3^2\) and \(5\) but nothing more . insuff

1) and 2) for any two positive integers X and Y, \(X*Y\) \(=\) \((LCM of X and Y) x (GCF of X and Y)\). So you have: \(2*5\) from \(GCF\) and \(2^2\) \(3^2\) and \(5\) from\(LCM\). So you can calculate exactly the value of \(X*Y\)

I 'll wait from Bunuel if my explanation is correct _________________

Re: If x and y are positive integers, what is the value of xy? [#permalink]

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13 Jul 2014, 22:36

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If x and y are positive integers, what is the value of xy? [#permalink]

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09 Nov 2014, 00:22

hello I've been trying to wrap my head around all that gcd and lcm would tell me about the numbers. When actual numbers are given it is a little easier. But when not, it gets hard, for me (like this : If a and b are positive integers divisible by 6, is 6 the greatest common divisor of a and b? (1) a = 2b + 6 (2) a = 3b ------sorry about this.. i cant paste urls yet)

So I need some generalizations (so i can summarize finally )

-What do GCD and LCM tell us about the numbers? -Is it ever possible to know the numbers themselves when the LCM or GCD are given? -We found the product of numbers here. Can we find the numbers themselves in any case? Please help me, even if you know the answer to one of these questions. Also let me know if any of them don't make sense, and why. Thanks loads

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