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Manager  Joined: 14 Jun 2010
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If x and y are positive integers, what is the value of xy?  [#permalink]

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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

Originally posted by ctgmat123 on 14 Jun 2010, 17:50.
Last edited by Bunuel on 30 Jun 2012, 05:29, edited 2 times in total.
Edited the question and added the OA
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Re: Data Sufficiency problem  [#permalink]

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ctgmat123 wrote:
Can someone walk me through this one?

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.

Hope it helps.
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satishreddy wrote:
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

This is just a formula based.

( GCFof x and y )* (LCM of x and y)= x * y

This is valid for all the numbers.Hope this will do.

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GMAT 1: 750 Q49 V42 Re: Data Sufficiency problem  [#permalink]

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Nice, I didn't know GCF(X&Y)*LCM(X&Y)=X*Y

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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.

Hence, xy = LCM x GCF.

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

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

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I think answer should be C as the property is GCF (x,y) * LCM(x,y) = Product of the two numbers
Hence value of xyv= 10*180= 1800

To prove A as insufficient you can plug numberssay X = 10 , Y = 30 GCF =10 product = 300 Another case X = 10 Y = 10 Hence product = 100

To prove B as insufficient you can plug in numbers X= 90 Y = 180 , Another case X=1 Y = 180
Hence Insufficient

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

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

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X and Y are integers

$$XY= ?$$

Statement 1

GCF( greatest Common factor) of x and y = $$10$$

x = 10 and y = 10
GCF = 10
hence xy = 100

x = 10 and y = 30
GCF = 10
hence xy = 300

Clearly Not Sufficient

Statement 2

LCM( least common multiple) of x and y = $$180$$

x= 180 and y = 1
LCM = 180
hence xy= 180

x=180 and y = 2
LCM = 180
Hence xy = 360

Clearly not sufficient

Now Combining Statement 1 and 2

we Know that
HCF of x and y * Lcm of x and y = x* y

Therefore $$10 * 180 = 1800$$

Sufficient

So Answer = C
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GRE 1: Q169 V154 Re: If x and y are positive integers, what is the value of xy?  [#permalink]

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Given data-> x and y are positive integers.
We need the value of x*y

Statement 1->
GCD(x,y)=10
x=10
y=10
GCD=10
xy=100

x=10
y=20
GCD=10
xy=200

Hence not sufficient.

Statement 2->
LCM(x,y)=180
x=180
y=180
LCM=180
xy=180*180

x=1
y=180
LCM=180
xy=180

Hence not sufficient.

Combing the two statements->

For two positive integers => LCM(x,y)*GCD(x,y)=> xy
Hence xy=180*10=> 1800

Hence sufficient.

Hence C.

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

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quiet888 wrote:
Is x and y are positive integers, what is the value of xy?

(i) The greatest common factor of x and y is 10
(ii) The least common multiple of x and y is 180

We can solve this questions using 2 approach.
Shorter approach (1 min)

Statement 1: GCF = 10 , (x,y) = (10,20), (10, 30), (20,30), (30,50)...... and so many numbers which are co-prime after division by 10.
NOT SUFFICIENT

Statement 2: LCM = 180 , (x,y)= (2,180), (10, 180), (90, 4).... and others
NOT SUFFICIENT

Combined : GCF * LCM = x*y .. Its a formula..
SUFFICIENT

Longer approach (Time taking: 2 min )
Statement 1: GCF = 10 , (x,y) = (10,20), (10, 30), (20,30), (30,50)...... and so many numbers which are co-prime after division by 10.
NOT SUFFICIENT

Statement 2: LCM = 180 , (x,y)= (2,180), (10, 180), (90, 4).... and others
NOT SUFFICIENT

Combined : x, y = (10,180), (20,90)
xy = 1800
SUFFICIENT

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

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quiet888 wrote:
Is x and y are positive integers, what is the value of xy?

(i) The greatest common factor of x and y is 10
(ii) The least common multiple of x and y is 180

Target question: What is the value of xy?

Statement 1: The greatest common factor of x and y is 10
case a: x=10 and y=10. Here, xy=100
case b: x=10 and y=20. Here, xy=200
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The least common multiple of x and y is 180
case a: x=1 and y=180. Here, xy=180
case b: x=2 and y=180. Here, xy=360
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 & 2
There's a nice rule that says:
If x and y are positive integers, then (GCF of x and y)(LCM of x and y)=xy
(aside: whenever a question mentions the LCM and the GCF, be sure to consider the above rule)
Statement 1 says the GCF of x and y is 10
Statement 2 says the LCM of x and y is 180
So, from our handy rule, xy = (10)(180) = 1800
Since we can now answer the target question with certainty, the combined statements are SUFFICIENT

So, the answer is C

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

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Can someone explain to me why 20 & 90 don't work?

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

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krisbrown12 wrote:
Can someone explain to me why 20 & 90 don't work?

Thank you.

x and y could be 20 and 90. In this case too xy = 20*90 = 1800. The same answer.
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Re: If x and y are positive integers, what is the value of xy?  [#permalink]

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ctgmat123 wrote:
If x and y are positive integers, what is the value of xy?

(1) The greatest common factor of x and y is 10
x, y could = 10 in which case xy = 100
or x,y could = 10, 20 so xy = 200
Not sufficient because without knowing the remaining factors of x or y we can't say anything about the value of xy.

(2) The least common multiple of x and y is 180
180 = 2²*3²*5, so it could be that x = 5 and y = 2²*3² in which case xy = 180
or x = 5*2, y = 3²*2² in which case xy = 360
Not sufficient because without knowing if they share any factors of 180 we can't determine xy.

Together, we know 1) GCF = 10 and LCM = 180, so x & y share 2*5 as factors, and the other 2*3² is split amongst them in some manner... product of xy = GCF*LCM = 1800, sufficient. Re: If x and y are positive integers, what is the value of xy?   [#permalink] 30 Sep 2019, 16:27
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