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If s and t are both primes, how many positive divisors

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If s and t are both primes, how many positive divisors [#permalink] New post 06 Aug 2014, 14:03
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Given \((s^3)(t^3) = v^2\) .If s and t are both primes, how many positive divisors of v are greater than 1, if v is an integer?

(A) two

(B) three

(C) five

(D) six

(E) eight
[Reveal] Spoiler: OA

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Re: GRE Math Challenge #3 [#permalink] New post 06 Aug 2014, 23:58
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s,t = 2 and v = 8. Positive divisors greater than 1 = {2,4,8}.
Therefore, answer would be three (B)
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Re: GRE Math Challenge #3- Solve (s^3)(t^3) = v^2 [#permalink] New post 17 Jul 2018, 15:04
soumya1989 wrote:
Level 170 question

Given \((s^3)(t^3) = v^2\) .If s and t are both primes, how many positive divisors of v are greater than 1, if v is an integer?

(A) two

(B) three

(C) five

(D) six

(E) eight



Tough one. Nice question. Tricky.

Note :

if we add 1 to the exponents of each primes and multiply them we get the total number of factors including 1 and the number itself.

Total factor : (3+1) (3+1) = 16.

Remember that 16 is the factor of \(v^2\). So, v has 4 factors for sure as v's factors was squared.

So, v has 4 factors.

Now , it's stated in the question that we need to find out the number of divisors of v that are greater than 1. In our factor calculation we consider 1 too. So it's time to deduct now.

Total number of factors greater than 1 = 4-1 = 3.

The correct answer is B.
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Re: If s and t are both primes, how many positive divisors [#permalink] New post 14 Dec 2019, 09:13
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Bump for further discussion
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Re: If s and t are both primes, how many positive divisors [#permalink] New post 15 Dec 2019, 19:27
what does "bump for further discussion" mean?
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Re: If s and t are both primes, how many positive divisors [#permalink] New post 16 Dec 2019, 03:02
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When a question (usually a good one) is buried somewhere on the board but it is worth "bump for further discussion" brings up it among the most recent posts to the students' attention.

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Re: If s and t are both primes, how many positive divisors [#permalink] New post 16 Dec 2019, 04:58
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soumya1989 wrote:
Given \((s^3)(t^3) = v^2\) .If s and t are both primes, how many positive divisors of v are greater than 1, if v is an integer?

(A) two

(B) three

(C) five

(D) six

(E) eight


Given: V is an integer => \(\sqrt{S^3 * T^3}\) => must be an integer

but given S and T are prime numbers, then \(\sqrt{S^3 * T^3}\) cannot be an integer, unless and until both the primes are same.

for V to be an integer, S = T , \(V^2 = S^3 * S^3 = S^6\); \(\sqrt{ V^2 }\) = \(\sqrt{(S^3)^2}\)
=> \(V = S^3\)
=> number of factors of V = (3 + 1) = 4 (rules)
=> Number of factors of V greater than 1 = (4 - 1) = 3

#Reference GMATCLUB.

https://youtu.be/aUTz3b5kgfY
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Re: If s and t are both primes, how many positive divisors   [#permalink] 16 Dec 2019, 04:58
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