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Is the total number of divisors of \(x^3\) a multiple of the total number of divisors of \(y^2\) ?

1. x = 4 2. y = 6

(1) x=4, x^3=64=2^6 --> number of factors of 64=6+1=7.

Now, can y^2 have the number of factors which is factor of 7? Well may have or may not. Number of factors of a square is odd. So y^2 should have either 1 factor (1^2) or 7 (81^2 or 8^2), both are possible, BUT y^2 can have other odd number of factors say 3 (5^2) and 3 is not factor of 7. Not sufficient

(2) y=6, y^2=36=2^2*3^2 --> number of factors of 36=(2+1)*(2+1)=9.

Can x^3 have the number of factors which is multiple of 9 (9, 18, 27, ...)? Let's represent x as the product of prime factors, x^3=(a^p*b^q*c^r)^3=a^3p*b^3q*c^3r. The number of factors would be (3p+1)(3q+1)(3r+1) and this should be multiple of 9. BUT (3p+1)(3q+1)(3r+1) is not divisible by 3, hence it can not be multiple of 9. The answer is NO. Sufficient.

Well not to complicate x^3 has 3k+1 number of distinct factors (1, 4, 7, 10, ... odd or even number), but it's not divisible by 3, thus not divisible by 9.

Re: Factors/Divisors DS Question From GMATClub Challenge Set 2 [#permalink]

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22 Oct 2009, 03:31

Bunuel, that is an incredible explanation. The OA is C, but it looks like the correct answer is indeed B.

One request: Can you expand a little bit more on the following statement: "(3p+1)(3q+1)(3r+1) is not divisible by 3." I agree with this statement using products of remainder logic (i.e., remainders can be multiplied so the product of the expression above will have a remainder of 1 when divided by 3), but I am curious about how you reason that the expression is not divisible by 3.

zero is divisible by any number..so 9 may or may not be multiple of the divisors of x^3..

FedX the point is not about: "... x=0? zero is divisible by any number.."

If x=0, then x^3=0, and every integer is the factor of zero, which means that 0 (x^3) has infinite number of divisors. Now, we want to determine if the equation 9*k=infinity makes sense, in other words is 9 factor of infinity?

I thought about this when writing my previous post and referred to the definition of Wiki: "In general, it is said ... m is a factor of n... if there exists an integer k such that n = km", in our case 9*k=infinity. I can not think about any integer k which gives us infinity upon multiplying by 9.

But maybe someone wants to clear this out farther more... How GMAC considers infinity?

slingfox can you please provide the source of the question and the explanation given there. Thanks.
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Bunuel, that is an incredible explanation. The OA is C, but it looks like the correct answer is indeed B.

One request: Can you expand a little bit more on the following statement: "(3p+1)(3q+1)(3r+1) is not divisible by 3." I agree with this statement using products of remainder logic (i.e., remainders can be multiplied so the product of the expression above will have a remainder of 1 when divided by 3), but I am curious about how you reason that the expression is not divisible by 3.

Thanks for the math lesson!

Well first of all "(3p+1)(3q+1)(3r+1)" came from - x^3=(a^p*b^q*c^r)^3=a^3p*b^3q*c^3r. To count the number of factor of x^3 we should use formula for determining their number: in general number of factors of an integer x greater than 1, expressed as product of prime factors x=a^p*b^q*c^r is (q+1)(p+1)(r+1). We have x^3=(a^p*b^q*c^r)^3=a^3p*b^3q*c^3r so the number of factors of x^3 will be (3p+1)(3q+1)(3r+1), now three multiples none of which is divisible by 3, hence their product is not divisible by 3.

Acc3ss wrote:

Just a query ..

In GMAT quant, Is there any dfference b/w divisors and factors ? or are they same ?

As I know divisor and factor are synonyms.
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Bunuel, here is the official explanation provided by GmatClub (the analysis seems pretty lacking):

"To get the divisors of \(x\) and \(y\) , we need their respective values.

Statement (1) by itself is insufficient. We can only find the divisors of \(x\) .

Statement (2) by itself is insufficient. We can only find the divisors of \(y\) .

Statements (1) and (2) combined are sufficient. Combining the statements, we have the divisors of \(x\) and \(y\) .

The correct answer is C."

Well, apparently I'm wrong, though I can't see it from the explanation. As far as this is GmatClub question can we ask moderators or someone else from GmatClub to clear this problem.

Re: Factors/Divisors DS Question From GMATClub Challenge Set 2 [#permalink]

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23 Oct 2009, 11:11

Well, I picked "C". We only need to find out that whether the number of divisors of one number are multiple of another number. No need to go into mechanics and find out the solution.

If x & y are known, then we can find out the solution,, so "C".

Well, I picked "C". We only need to find out that whether the number of divisors of one number are multiple of another number. No need to go into mechanics and find out the solution.

If x & y are known, then we can find out the solution,, so "C".

Correct me if I am wrong.

In DS you should determine which statement is enough to answer the question asked. It's obvious that knowing two number (statement 1 + statement 2) is sufficient to answer the question, but I claim that the answer can be determined based ONLY on statement 2. If I'm right then according to the idea of Data Sufficiency the answer must be D - statement 2 is sufficient to answer the question alone.

P.S. Q is: Is the total number of divisors of x^3 a multiple of the total number of divisors of y^2?
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I don't care for the question at all, since it is not the type of divisibility question you would ever see on the GMAT. Every GMAT divisibility question will tell you in advance that any unknowns represent positive integers. If, in the above question, we knew that x and y were positive integers, then Bunuel's solution is correct, and the answer should be B. If, however, we do not know in advance that x is an integer, then it's certainly possible that x is equal to the cube root of 36, and then using Statement 2 alone, x^3 and y^2 are equal, and it's possible that the answer to the question is 'yes'. So because of this technicality, Statement 2 is not sufficient alone. That's the type of technicality that is never tested on the GMAT, however.

I'd add, in reference to the discussion about infinity above, that all numbers on the GMAT are real numbers. Infinity is not a real number, so you don't need to worry about questions like 'is infinity a multiple of 9' (that question doesn't make any mathematical sense anyway).
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I don't care for the question at all, since it is not the type of divisibility question you would ever see on the GMAT. Every GMAT divisibility question will tell you in advance that any unknowns represent positive integers. If, in the above question, we knew that x and y were positive integers, then Bunuel's solution is correct, and the answer should be B. If, however, we do not know in advance that x is an integer, then it's certainly possible that x is equal to the cube root of 36, and then using Statement 2 alone, x^3 and y^2 are equal, and it's possible that the answer to the question is 'yes'. So because of this technicality, Statement 2 is not sufficient alone. That's the type of technicality that is never tested on the GMAT, however.

I'd add, in reference to the discussion about infinity above, that all numbers on the GMAT are real numbers. Infinity is not a real number, so you don't need to worry about questions like 'is infinity a multiple of 9' (that question doesn't make any mathematical sense anyway).

Thanks again Ian for response.

Here I want to make some remarks:

You stated: "If, however, we do not know in advance that x is an integer, then it's certainly possible that x is equal to the cube root of 36, and then using Statement 2 alone, x^3 and y^2 are equal, and it's possible that the answer to the question is 'yes'. "

I agree that GMAT would stated that we are dealing with positive integers, but mathematically we are not asked whether x^3=y^2, but whether the total number of distinct factors of y^2, which in this case is 9 (for statement 2), can be the factor of the total number of distinct factors of x^3.

As for infinity: yes, GMAT won't test this and yes 9 multiple of infinity doesn't make sense, that's what I wanted to state. I had to consider this case as when x=0 x^3=0, thus x^3 has infinite number of multiples. So, I tried to demonstrate that even in this case 9 can not be the multiple of factors of x^3, as 9 multiple of infinity makes no sense.

Well, if x^3 = y^2, then the number of factors of x^3 is equal to the number of factors of y^2, so of course the number of divisors of x^3 is a multiple of the number of divisors of y^2. So, if we allow x to be a non-integer, it is possible that the answer to the question is yes, using only Statement 2. However, if x must be an integer, using Statement 2 alone, the answer to the question must be 'no', as you demonstrated earlier.

Bunuel wrote:

What I tried to show in my first post is that, no matter x and y are integers or not, it's impossible to be true when number of multiples of y^2 is 9.

In your solution, you assumed x had a prime factorization. That is, you assumed x was a positive integer greater than 1. Your solution does not apply if x is a non-integer.
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That's what I wanted to find out: whether my logic of determining the answer was correct. If it would be the real GMAT question my way would be applicable. In original question as x may not be the integer then factorization is not right.

Every GMAT divisibility question will tell you in advance that any unknowns represent positive integers.

If it were the real GMAT question the solution in my first post is right.

Answer: B.

As it's not stated that x and y are integers, then x^3 may be equal to y^2 and thus will have the equal number of distinct factors as y^2. Hence in that case my solution is not applicable.

So, the answer to the original question: C.
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Re: Factors/Divisors DS Question From GMATClub Challenge Set 2 [#permalink]

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24 Oct 2009, 23:24

Bunuel wrote:

Conclusion:

Every GMAT divisibility question will tell you in advance that any unknowns represent positive integers.

If it were the real GMAT question the solution in my first post is right.

Answer: B.

As it's not stated that x and y are integers, then x^3 may be equal to y^2 and thus will have the equal number of distinct factors as y^2. Hence in that case my solution is not applicable.

So, the answer to the original question: C.

Bunuel, that was a good discussion.

However I am not sure about your statement that:

Every GMAT divisibility question will tell you in advance that any unknowns represent positive integers.

If that were the case, a given question in divisibility will be lot more easier. How did you conclude this? Can you verify it again?
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Every GMAT divisibility question will tell you in advance that any unknowns represent positive integers.

If it were the real GMAT question the solution in my first post is right.

Answer: B.

As it's not stated that x and y are integers, then x^3 may be equal to y^2 and thus will have the equal number of distinct factors as y^2. Hence in that case my solution is not applicable.

So, the answer to the original question: C.

Bunuel, that was a good discussion.

However I am not sure about your statement that:

Every GMAT divisibility question will tell you in advance that any unknowns represent positive integers.

If that were the case, a given question in divisibility will be lot more easier. How did you conclude this? Can you verify it again?

I was not sure about it either. As we are dealing with the questions from various sources, sometimes not really good ones, I had many questions which were not stating that. But I refer to Ian's post who is GMAT instructor thus must know how GMAT works much better then I do.

Re: Factors/Divisors DS Question From GMATClub Challenge Set 2 [#permalink]

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25 Oct 2009, 00:11

Bunuel wrote:

GMAT TIGER wrote:

Bunuel wrote:

Conclusion:

Every GMAT divisibility question will tell you in advance that any unknowns represent positive integers.

If it were the real GMAT question the solution in my first post is right.

Answer: B.

As it's not stated that x and y are integers, then x^3 may be equal to y^2 and thus will have the equal number of distinct factors as y^2. Hence in that case my solution is not applicable.

So, the answer to the original question: C.

Bunuel, that was a good discussion.

However I am not sure about your statement that:

Every GMAT divisibility question will tell you in advance that any unknowns represent positive integers.

If that were the case, a given question in divisibility will be lot more easier. How did you conclude this? Can you verify it again?

I was not sure about it either. As we are dealing with the questions from various sources, sometimes not really good ones, I had many questions which were not stating that. But I refer to Ian's post who is GMAT instructor thus must know how GMAT works much better then I do.

(Please see Ian's first post)

Oh ok...

Lets wait for his/her clarification...
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