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If x and y are positive integers, is xy a multiple of 8?

a) Greatest common divisor of x and y is 10 b) Least common factor of x and y is 100

(1) \(GCD(x,y)=10\) --> if \(x=10\) and \(y=10\), then \(xy=100\), which is not divisible by 8 BUT if \(x=10\) and \(y=20\), then \(xy=200\) which is divisible by 8. Two different answers. Not sufficient.

(2) \(LCM(x,y)=100\) --> if \(x=1\) and \(y=100\), then \(xy=100\), which is not divisible by 8 BUT if \(x=4\) and \(y=50\), then \(xy=200\) which is divisible by 8. Two different answers. Not sufficient.

(1)+(2) The most important property of LCM and GCD is: for any positive integers \(x\) and \(y\), \(x*y=GCD(x,y)*LCM(x,y)=10*100=1000\) --> 1000 is divisible by 8. Sufficient.

Stmnt 1: GCD = 10 If GCD is 10, LCM will be a multiple of 10 (the link above will explain why). Let us say LCM = 10a x*y = 10*10a = 100a We still cannot say whether xy is divisible by 8. Not sufficient.

Stmnt2: LCM = 100 If LCM is 100, we cannot say what GCD is. It could be 1 or 10 or 50 etc. x*y = GCD*100 We cannot say whether xy is divisible by 8. Not sufficient.

Taking both stmnts together, x*y = 10*100 = 1000 Since 1000 is divisible by 8, x*y is divisible by 8. Sufficient. Answer (C).
_________________

Stmnt 1: GCD = 10 If GCD is 10, LCM will be a multiple of 10 (the link above will explain why). Let us say LCM = 10a x*y = 10*10a = 100a We still cannot say whether xy is divisible by 8. Not sufficient.

Stmnt2: LCM = 100 If LCM is 100, we cannot say what GCD is. It could be 1 or 10 or 50 etc. x*y = GCD*100 We cannot say whether xy is divisible by 8. Not sufficient.

Taking both stmnts together, x*y = 10*100 = 1000 Since 1000 is divisible by 8, x*y is divisible by 8. Sufficient. Answer (C).

Hi Karishma,

I cehcked the link but I did not understand why LCM will be a multiple of 10.. I understand that LCM*GCD=prod of 2 nos.. but don't understand why LCM has to be a multiple of 10..

Also, could you please help us with an approach on generating numbers for testing when a LCM is given.. in this case, lcm of x and y is 100.. .so how do we generate numbers whose lcm would be 100.

For gcd, we can write the GCD and multiply that by different numbers to get the pair of numbers.. Kindly explain how to do that for LCM
_________________

hope is a good thing, maybe the best of things. And no good thing ever dies.

I cehcked the link but I did not understand why LCM will be a multiple of 10.. I understand that LCM*GCD=prod of 2 nos.. but don't understand why LCM has to be a multiple of 10..

Also, could you please help us with an approach on generating numbers for testing when a LCM is given.. in this case, lcm of x and y is 100.. .so how do we generate numbers whose lcm would be 100.

For gcd, we can write the GCD and multiply that by different numbers to get the pair of numbers.. Kindly explain how to do that for LCM

GIven that GCD = 10. What is GCD? It is the greatest common factor of two numbers i.e. both the numbers must have that factor. When you find the LCM of the numbers, the LCM includes all the factors of both the numbers. Hence, it will include 10 too. e.g. GCD = 10 Numbers: 10x, 10y where x and y are co-prime. What will be the LCM? LCM = 10xy (includes all factors of both the numbers)

In this question you don't need to list out the possible numbers given LCM = 100 but if you need to do it in another question, this is how you can handle that:

LCM = 100 = 2^2*5^2 Numbers: Split the primes -> (4, 25) Make one number = LCM -> (1, 100), (2, 100), (4, 100), (5, 100), (10, 100), (20, 100), (25, 100), (50, 100), (100, 100) One number must have the highest power of each prime -> (2^2*5, 2*5^2 which is 20, 50), (2^2, 2*5^2 which is 4, 50), (2^2*5, 5^2 which is 20, 25)

The overall strategy is this: Split the LCM into its prime factors. At least one number must have the highest power of each prime.

LCM = 2^a*3^b*5^c

At least one number must have 2^a, same or another number must have 3^b and same or another number must have 5^c. There are various possibilities.
_________________

Stmnt 1: GCD = 10 If GCD is 10, LCM will be a multiple of 10 (the link above will explain why). Let us say LCM = 10a x*y = 10*10a = 100a We still cannot say whether xy is divisible by 8. Not sufficient.

Stmnt2: LCM = 100 If LCM is 100, we cannot say what GCD is. It could be 1 or 10 or 50 etc. x*y = GCD*100 We cannot say whether xy is divisible by 8. Not sufficient.

Taking both stmnts together, x*y = 10*100 = 1000 Since 1000 is divisible by 8, x*y is divisible by 8. Sufficient. Answer (C).

Hi Karishma,

I cehcked the link but I did not understand why LCM will be a multiple of 10.. I understand that LCM*GCD=prod of 2 nos.. but don't understand why LCM has to be a multiple of 10..

Also, could you please help us with an approach on generating numbers for testing when a LCM is given.. in this case, lcm of x and y is 100.. .so how do we generate numbers whose lcm would be 100.

For gcd, we can write the GCD and multiply that by different numbers to get the pair of numbers.. Kindly explain how to do that for LCM

I will look at this in a simpler manner.

GCF is the multiple of the lowest power of common factors (of x & y). LCM is the multiple of highest power of common factors. Therefore GCF is a factor is LCM. Since statement 1 says GCF is 10, so LCM can be assumed as multiple of 10.

Re: If x and y are positive integers, is xy a multiple of 8? (1) [#permalink]

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06 Jul 2013, 20:54

For xy to be a multiple of 8 , xy should have minimum of 3 2's.

1. Gcd(x,y)=10 ===> x= 5*2 *m (where m can be any integer) y= 5*2* n (where n can be any integer) This does not clearly say that xy will be divisible by 8 because we are sure of only two 2's.

2. Lcm(x,y)= 100= 5^2* 2^2.====> It gives us different combinations of x and y. For eg: x= 5^2*2^2, y= 5*1, or x=5* 2^2, y= 5^2* 2 etc. Hence, xy may or may not be divisible by 8.

When 1+ 2 then, x= 5*2 , y= 5^2* 2^2 or vice versa. In any case we can see that xy is a multiple of 8.

Re: If x and y are positive integers, is xy a multiple of 8? (1) [#permalink]

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17 Sep 2015, 08:11

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Re: If x and y are positive integers, is xy a multiple of 8? (1) [#permalink]

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02 Oct 2016, 09:27

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
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Re: If x and y are positive integers, is xy a multiple of 8? (1) [#permalink]

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07 Apr 2017, 16:32

I believe these sort of questions can be handled without remembering the formula : X*Y = LCM (X, Y) * GCF (X,Y)

Let's see how,

Statement 1 : The greatest common divisor of x and y is 10

What this essentially means is that we have two numbers x and y, that can be written in the following way : x = 2*5*A y = 2*5*B , where A and B are co-primes. (If they had any common factor, it would've been considered in the GCF) Now, x*y = 2*5*2*5*A*B => Now in order for x*y to be a factor of 8, we need at least one more 2 in either A or B; A condition can cannot be guaranteed just with the statement 1. Hence, insufficient.

Statement 2 : The least common multiple of x and y is 100

Statement 2 alone is not sufficient. Consider the below 2 examples :

Let x = 1 and y = 100. LCM (x,y) = 100 and x*y = 100 (not a multiple of 8) Let x = 2 and y = 2*2*5*5. LCM (x,y) = 100 and x*y = 200 (multiple of 8)

Hence, statement 2 alone is not sufficient.

Now, considering statement 1 and 2 together :

Statement 1 : The greatest common divisor of x and y is 10 x = 2*5*A y = 2*5*B , where A and B are co-primes.

Statement 2 : The least common multiple of x and y is 100 LCM (x,y) = 2*5 *A*B = 100 => A*B = 10 => The only combination possible is 1*10 since A and B are co-primes.

Now , x*y = 2*5*2*5*2*5 (Always a multiple of 8).

Option C is the correct answer. Would like to know your thoughts on using this approach while solving similar problems.

Re: If x and y are positive integers, is xy a multiple of 8? (1) [#permalink]

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29 Apr 2017, 17:30

Nice official question. I made test cases to reject Statement 1 and 2 as => Statement 1 =>(10,10) and (10,10000) => Insufficient Statement 2 =>(1,100) and (100,100)=> Insufficient

Combing them and using LCM*GCD=Product of two integers => xy=1000=8*125 Hence xy is a multiple of 8