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

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.

Answer: C.

Hope it helps.
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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.

Consider KUDOS if u find it helpful.Thanks.

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

Kudos!!

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.

--
MBA Gambit: NYC GMAT Tutor
https://www.mbagambit.com

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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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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Consider Kudos,If My post helped!!!!!!!!!!!!!!!!!!!!!!!!!!!!

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

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

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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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


Answer C

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

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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(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.

Solution:

Statement One Only:

The greatest common factor of x and y is 10.

If x = 10 and y = 20, then xy = 200. However, if x = 10 and y = 30, then xy = 300. Statement one alone is not sufficient.

Statement Two Only:

The least common multiple of x and y is 180.

If x = 1 and y = 180, then xy = 180. However, if x = 60 and y = 90, then xy = 5400. Statement two alone is not sufficient.

Statements One and Two Together:

Recall that the product of the LCM and GCF of two positive integers is the product of the two integers. We have xy = 10 * 180 = 1800. The two statements together are sufficient.

Answer: C
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Forget the conventional way to solve DS questions.

We will solve this DS question using the variable approach.

DS question with 2 variables: Let the original condition in a DS question contain 2 variables. Now, 2 variables would generally require 2 equation for us to be able to solve for the value of the variable.

We know that each condition would usually give us an equation, and Since we need 2 equations to match the numbers of variables and equations in the original condition, the logical answer is C.

To master the Variable Approach, visit https://www.mathrevolution.com and check our lessons and proven techniques to score high in DS questions.

Let’s apply the 3 steps suggested previously. [Watch lessons on our website to master these 3 steps]

Step 1 of the Variable Approach: Modifying and rechecking the original condition and the question. We have to find the value of 'xy'.

=> Given that 'x' and 'y' are positive integers

Second and the third step of Variable Approach: From the original condition, we have 2 variables (x and y).To match the number of variables with the number of equations, we need 2 equations. Since conditions (1) and (2) will provide 2 equations, C would most likely be the answer.

Let’s take look at both conditions together.

Condition(1) tells us that the greatest common factor of x and y is 10 .

Condition(2) tells us that The least common multiple of x and y is 180 .


=> Product of numbers = product of their GCF and LCM

=> xy = 180 * 10 = 1800

Since the answer is unique, both conditions combined together are sufficient by CMT 2.


Both conditions combined together are sufficient.

So, C is the correct answer.

Answer: C

Solution:

Statement One Only:

The greatest common factor of x and y is 10.

If x = 10 and y = 20, then xy = 200. However, if x = 10 and y = 30, then xy = 300. Statement one alone is not sufficient.

Statement Two Only:

The least common multiple of x and y is 180.

If x = 1 and y = 180, then xy = 180. However, if x = 60 and y = 90, then xy = 5400. Statement two alone is not sufficient.

Statements One and Two Together:

Recall that the product of the LCM and GCF of two positive integers is the product of the two integers. We have xy = 10 * 180 = 1800. The two statements together are sufficient.

Answer: C
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This is a very simple question on the concepts of HCF and LCM. From the question data, we know that x and y are positive integers. We need to find the value of xy.

From statement I alone, HCF(x,y) = 10. This is insufficient since x and y can take a host of values such that their HCF is 10.

For example, x = 10 and y = 10 in which case their HCF is 10 and their product is 100.
x = 10 and y = 20 in which case their HCF is still 10 but their product is 200.

Statement I alone is insufficient to give us a unique value for xy. Answer options A and D can be eliminated. Possible answer options are B, C or E.

From statement II alone, LCM(x,y) = 180. This is insufficient since x and y can take a host of values such that their LCM is 180.

For example, x = 36 and y = 5 in which case their LCM is 180 and their product is 180.
x = 90 and y = 20 in which case their LCM is 180 but their product is 1800.

A common mistake that some test takers make here is to remember the data from the first statement and therefore conclude that the first case is not possible. This is not right, note that you are trying to solve the question using the second statement alone.

Statement II alone is insufficient. Answer option B can be eliminated. Possible answer options are C or E.

Combining statements I and II, we have the following:
From statement I, HCF (x,y) = 10; from statement II alone, LCM (x,y) = 180.

Combining these two pieces of information should be done using a property rather than plugging in values. Remember, you are not trying to find the values of x and y individually; you are trying to calculate xy.

For any two numbers, Product of numbers = Product of their LCM and HCF.

Therefore, xy = 180 * 10 = 1800.
The combination of statements is sufficient to find a unique value of xy. Answer option E can be eliminated.

The correct answer option is C.

Hope that helps!
Aravind B T
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