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If x is a prime number, what is the number of factors of 75x?

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Bunuel
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If x is a prime number, what is the number of factors of 75x? [#permalink] New post   22 May 2023, 09:30
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Question Stats:

49% (01:45) correct 51% (01:39) wrong based on 78 sessions
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If x is a prime number, what is the number of factors of 75x?

(1) x^2 has 3 factors
(2) x > 5
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B
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swapnils123
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Re: If x is a prime number, what is the number of factors of 75x? [#permalink] New post   25 May 2023, 00:19
Bunuel is the question correct?
I can find no of factors even without statements 1 and 2.
See this: If x is a prime number, then we can write: 75x = 3 * 5^2 * x which is 3^1 * 5^2 * x^1. So no of factors = (1+1) * (2+1) * (1+1) = 12.

Correct me if I am wrong.
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Re: If x is a prime number, what is the number of factors of 75x? [#permalink] New post   25 May 2023, 00:26
Expert Reply
swapnils123 wrote:
Bunuel is the question correct?
I can find no of factors even without statements 1 and 2.
See this: If x is a prime number, then we can write: 75x = 3 * 5^2 * x which is 3^1 * 5^2 * x^1. So no of factors = (1+1) * (2+1) * (1+1) = 12.

Correct me if I am wrong.


How do you know that x is not 3 or 5?
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Re: If x is a prime number, what is the number of factors of 75x? [#permalink] New post   25 May 2023, 00:31
hmm ..okay. I see my mistake. Thanks for clarifying.
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Re: If x is a prime number, what is the number of factors of 75x? [#permalink] New post   25 May 2023, 00:57
Only when prime no 2 is involved..

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Re: If x is a prime number, what is the number of factors of 75x? [#permalink] New post   07 Nov 2023, 05:23
Good one.. I got it wrong but quickly realized where I went wrong..

Answer is B

If x is a prime number, what is the number of factors of 75x?

(1) x^2 has 3 factors
(2) x > 5

Question - 75 has 5.5.3.x
We know - x = 2,3,5,7,11.... (prime numbers ONLY)

1. Square any number from set X.. we get many squares but we do not know if X is 3 or 5..
IF X = 3, then we get number of factors as 3.3 = 9
If X = 7, then we get number of factors as 3.2.2 = 12
Hence insufficient

2. Since X is greater than 5, any prime number can be plugged in with power 1 and we will get # of factors as 3.2.2 = 12
Hence sufficient
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Re: If x is a prime number, what is the number of factors of 75x? [#permalink] New post   09 Nov 2023, 08:45
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Bunuel wrote:
If x is a prime number, what is the number of factors of 75x?
(1) x^2 has 3 factors
(2) x > 5


Solution:
Pre Analysis:
  • Given that x is a prime number
  • \(75x=3\times 5^2\times x\)
  • 3 cases possible here:
    • Case 1: If \(x=3\), then \(75x=3^2\times 5^2\). Ann the number of factors \(=(2+1)\times (2+1)=9\)
    • Case 2: If \(x=5\), then \(75x=3\times 5^3\). Ann the number of factors \(=(1+1)\times (3+1)=8\)
    • Case 3: If \(x=\) any other prime, then \(75x=3\times 5^2\times x^1\). Ann the number of factors \(=(1+1)\times (2+1)\times (1+1)=12\)

Statement 1: \(x^2\) has 3 factors
  • We know that x is a prime number
  • So, \(x^2\) will obviously have \((2+1)=3\) factors
  • Thus, statement 1 alone is not only insufficient but also redundant and we can eliminate options A, C and D

Statement 2: \(x > 5\)
  • This aligns with case 3 of our pre analysis step.
  • So, total factors \(=12\)
  • Thus, statement 2 alone is sufficient


Hence the right answer is Option B
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Re: If x is a prime number, what is the number of factors of 75x? [#permalink]
09 Nov 2023, 08:45
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