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Re: If n = x^5*y^7, where x and y are positive integers greater than 1 [#permalink]

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08 Aug 2016, 06:38

MarkusKarl wrote:

I'll give it a try.

A) X, Y = prime.

If X =/= Y then we would know the answer (6*8 factors). However, if X=Y then we would receive a lot of answers that would generate the same factors with the previous method. e.g. x=y=3 => 3^3*3^1=3^1*3^3.

Thus, A is insufficient.

As for B) (Which, in all honesty, was my first choice before I analyzed the question further). We now know that x and y consist of 2 prime numbers.

IF x and y are two different prime numbers we would be able to answer the question. However(and here was my mistake), that information is given in (A). (In that case it would indeed be 6*8 factors).

X and Y can still be different composite numbers without giving n more than 2 prime factors. e.g. x = 2*3 and y = 2*2*3. X and Y can also be composite numbers where x and y are two different prime numbers taken to any power.

Thus, (as written above) We needed the information about x and y being prime numbers from (A) and we needed the information that n only had two different prime factors from (B). Thus -> C is the answer.

Great question, thanks!

Can a single prime number work for both X and Y? If so then it should be written X and X, I think, not X and Y.

Re: If n = x^5*y^7, where x and y are positive integers greater than 1 [#permalink]

Show Tags

08 Aug 2016, 07:21

NaeemHasan wrote:

MarkusKarl wrote:

I'll give it a try.

A) X, Y = prime.

If X =/= Y then we would know the answer (6*8 factors). However, if X=Y then we would receive a lot of answers that would generate the same factors with the previous method. e.g. x=y=3 => 3^3*3^1=3^1*3^3.

Thus, A is insufficient.

As for B) (Which, in all honesty, was my first choice before I analyzed the question further). We now know that x and y consist of 2 prime numbers.

IF x and y are two different prime numbers we would be able to answer the question. However(and here was my mistake), that information is given in (A). (In that case it would indeed be 6*8 factors).

X and Y can still be different composite numbers without giving n more than 2 prime factors. e.g. x = 2*3 and y = 2*2*3. X and Y can also be composite numbers where x and y are two different prime numbers taken to any power.

Thus, (as written above) We needed the information about x and y being prime numbers from (A) and we needed the information that n only had two different prime factors from (B). Thus -> C is the answer.

Great question, thanks!

Can a single prime number work for both X and Y? If so then it should be written X and X, I think, not X and Y.

Hi,

I am not sure that I understand your question, but i will try to respond.

The question does not state that x and y are different numbers. As such, we are not able to answer the question with just a. We receive that information when a and b are combined.

Please let me know if I misunderstood the question or if I can clarify further.

Best wishes

Posted from my mobile device _________________

I love being wrong. An incorrect answer offers an extraordinary opportunity to improve.

Re: If n = x^5*y^7, where x and y are positive integers greater than 1 [#permalink]

Show Tags

08 Aug 2016, 09:52

If x and y are positive integers and n=x5∗y7n=x5∗y7, then how many positive divisors does n have?

(1) x does not have a factor p such that 1 < p < x and y does not have a factor q such that 1 < q < y. (2) n has only two prime factors.

Giving it a try (This was actually my second attempt):

S1 states that X and Y are prime numbers with no current constraints

Can we find the number of positive divisors of N from that? (MGMAT says to work to prove the statement insuff)

Scen 1: (X = 2, Y = 3) (2^5)(3^7) No of divisors (5+1)(7+1) = 48

Scen 2 (X = 2, Y = 2) (2^12) No. of Divisors =/= 48

S1 is insufficient due to multiple answers

S2 states that N has two different prime numbers meaning that X and/or Y can be any numbers that share at most two prime factors (eg: [2,3],[2,6], [2,10], [5,15], [15,15])

Scen 1: (X = 2, Y = 3) (2^5)(3^7) No of divisors (5+1)(7+1) = 48

Scen 2 (X = 2, Y = 6) (PF = 2 & 3) Honestly, I don't even know what the number of divisors would be but I know it's not 48, and I'm not going to waste time trying to figure out what that number is. Knowing it's not 48 is enough for me to say it's insufficient.

So A, D, & B are eliminated

Now to try S1 & S2 together:

S1: X and Y are prime numbers & S2: X and Y must be different prime numbers

Scen 1: (X = 2, Y = 3) (2^5)(3^7) No of divisors (5+1)(7+1) = 48

Scen 1: (X = 2, Y = 5) (2^5)(5^7) No of divisors (5+1)(7+1) = 48

During my first try while I was scribbling on paper I read S2 as having the parameters of S1 & S2 combined. I essentially read it as X and Y have to be different prime numbers, which was the incorrect way to read it.

Re: If n = x^5*y^7, where x and y are positive integers greater than 1 [#permalink]

Show Tags

09 Aug 2016, 02:35

MarkusKarl wrote:

NaeemHasan wrote:

MarkusKarl wrote:

I'll give it a try.

A) X, Y = prime.

If X =/= Y then we would know the answer (6*8 factors). However, if X=Y then we would receive a lot of answers that would generate the same factors with the previous method. e.g. x=y=3 => 3^3*3^1=3^1*3^3.

Thus, A is insufficient.

As for B) (Which, in all honesty, was my first choice before I analyzed the question further). We now know that x and y consist of 2 prime numbers.

IF x and y are two different prime numbers we would be able to answer the question. However(and here was my mistake), that information is given in (A). (In that case it would indeed be 6*8 factors).

X and Y can still be different composite numbers without giving n more than 2 prime factors. e.g. x = 2*3 and y = 2*2*3. X and Y can also be composite numbers where x and y are two different prime numbers taken to any power.

Thus, (as written above) We needed the information about x and y being prime numbers from (A) and we needed the information that n only had two different prime factors from (B). Thus -> C is the answer.

Great question, thanks!

Can a single prime number work for both X and Y? If so then it should be written X and X, I think, not X and Y.

Hi,

I am not sure that I understand your question, but i will try to respond.

The question does not state that x and y are different numbers. As such, we are not able to answer the question with just a. We receive that information when a and b are combined.

Please let me know if I misunderstood the question or if I can clarify further.

Best wishes

Posted from my mobile device

My point is that X and Y can not represent a sigle value. As X and Y are different letters they should reflect different values and hence A should be the answer. Hope, you got my point now.

Re: If n = x^5*y^7, where x and y are positive integers greater than 1 [#permalink]

Show Tags

09 Aug 2016, 08:10

NaeemHasan wrote:

My point is that X and Y can not represent a sigle value. As X and Y are different letters they should reflect different values and hence A should be the answer. Hope, you got my point now.

X and Y are different variables that represent some integer value, and without specific constraints they each can represent any value on the number line: -∞ < X < ∞ & -∞ < Y < ∞. This also means that, unless otherwise stated, X = Y is a possibility.

My point is that X and Y can not represent a sigle value. As X and Y are different letters they should reflect different values and hence A should be the answer. Hope, you got my point now.

Unless it is explicitly stated otherwise, different variables CAN represent the same number.
_________________

Re: If n = x^5*y^7, where x and y are positive integers greater than 1 [#permalink]

Show Tags

09 Aug 2016, 08:56

Bunuel wrote:

NaeemHasan wrote:

My point is that X and Y can not represent a sigle value. As X and Y are different letters they should reflect different values and hence A should be the answer. Hope, you got my point now.

Unless it is explicitly stated otherwise, different variables CAN represent the same number.

Re: If n = x^5*y^7, where x and y are positive integers greater than 1 [#permalink]

Show Tags

21 Aug 2016, 03:47

NaeemHasan wrote:

Statement 1 is sufficient alone. If we know that X and Y are primes then obviously we can determine the number of factors of n. Total factors are (5+1)*(7+1)=48. But unfortunately the OA is C. Where is my mistake? Can you, bunuel, help me to understand the fact, please.

Hi there See if x=y => number of factors = 13 if not then the number of factors = 48 statement 2 is insufficient as x=2*3 and y=1 is different from x=3 and y=2 combining them we can say that x and y are both different primes hence the number of factors = 48 Smash that C
_________________

Hi..I got the correct answer..but I have a question.

Consider statement

(1) x does not have a factor p such that 1 < p < x and y does not have a factor q such that 1 < q < y.

Does this statement stop any of x and y from being = 1 ??

If we translate it in mathematical terms..it will become 1<p<1. So x = 1 is possible because nothing satisfies the translated condition..maybe I'm just being silly but it still is intriguing to me
_________________

Hi..I got the correct answer..but I have a question.

Consider statement

(1) x does not have a factor p such that 1 < p < x and y does not have a factor q such that 1 < q < y.

Does this statement stop any of x and y from being = 1 ??

If we translate it in mathematical terms..it will become 1<p<1. So x = 1 is possible because nothing satisfies the translated condition..maybe I'm just being silly but it still is intriguing to me

1 < p < x means that x is greater than 1. 1 < q < y means that y is greater than 1.
_________________

If \(n = x^5*y^7\), where \(x\) and \(y\) are positive integers greater than 1, then how many positive divisors does \(n\) have?

(1) \(x\) does not have a factor \(p\) such that \(1 < p < x\) and \(y\) does not have a factor \(q\) such that \(1 < q < y\). This statement implies that both \(x\) and \(y\) are primes (a prime number does not have a factor which is greater than 1 and less than itself, it has only two factors 1 and itself). Now, if \(x\) and \(y\) are different primes, then the number of factors of \(n\) will be \((5+1)(7+1)=48\) but if \(x=y\), then \(n = x^5*y^7=x^{12}\) and it will have 13 factors. Not sufficient.

(2) \(n\) has only two prime factors. If those primes are \(x\) and \(y\), then the number of factors of \(n\) will be \((5+1)(7+1)=48\) but if, say \(x=2*3=6\) and \(y=2\), then \(n = x^5*y^7=2^{12}*3^5\) and it will have \((12+1)(5+1)=78\) factors. Not sufficient.

(1)+(2) Since from (2) \(n\) has only two prime factors, then from (1) it follows that \(x\) and \(y\) are different primes so the number of factors of \(n\) will be \((5+1)(7+1)=48\). Sufficient.