If n = x^5*y^7, where x and y are positive integers greater than 1

The answer cannot be anything but D.
First statement clearly means that both X & Y are prime . They can be equal as well as two separate prime numbers. If equal then no of factors would be 13 , if different then it would be 48.
So insufficient .

The second statement says that n has only 2 prime factors so X can be 2,4,8 and Y can be 3,9,27 ..all yielding different factors for n.
Insufficient .

But if we combine we surely know X & Y are prime and since n has only 2 prime factors both X & Y are different .
Sufficient .


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Bunuel, may I have some assistance with where I'm going wrong with (1+2)?

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

\(x\) and \(y\) are either prime or 1. Assume both are prime for a counterexample.

\(x = y = 2 \implies n = 2^{12} \text{ (}12 + 1 \text{ divisors})\\\\
x = 2, y = 3 \implies n = 2^53^7 ((5+1)(7+1) = 48 \text{ divisors})\)

Insufficient

(2)\(n\) has only two prime factors.

Assuming this means two distinct prime factors, due to the integer constraint on \(x\) and \(y\), I do not believe that two prime factors is possible (setting x and y to 1 would fail, as would 1 and any other integer).

\(x = 2, y = 3 \implies n = 2^53^7 ((5+1)(7+1) = 48 \text{ divisors})\\\\
x = 6, y = 2 \implies n = 3^52^{12} ((5+1)(12+1) = 72 \text{ divisors})\)

Insufficient

(1 + 2)

(I can't for the life of me determine why this is not sufficient, none of the above answers seem correct to me).

x and y must be positive integers (question stem)
x or y may not be 1. (1) requires x and y are prime or 1, (2) requires that there are 2 prime factors, so we need two primes.
x and y must therefore be prime (1)
x and y must be distinct primes (2)

\(x^k \times y^j\) will always have \((k+1)(j+1)\) divisors when \(x\) and \(y\) are distinct primes.

Therefore sufficient.

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

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

So the answer is C.

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.

Bunuel

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.

Easy to miss something here. Great question!

For (1), we can not rule out the possiblity of x = y.

For (2), n could have only 1 prime factor and we still would not be able to solve for the total number of factors.

If x = 343, y = 49 --> n = 7^32, Factors = 33.
If x = 7, y = 7 --> n = 7^12, Factors = 13.

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