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Kem12
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How many different prime factors does positive integer n have? 30 Jun 2018, 08:54
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Bunuel
Kem12
I agree with all the explanations but have a question though... If n can be 7,8 or 9, which isn't a definite value, does that make the answer sufficient?

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The question does not ask "what is the value of n", it asks "How many different prime factors does positive integer n have". So, for (1), even though n can take three values, each of those values still has only one prime factor. For (2) we get that n is a prime number, so it must have only one prime factor. Thus, both statements give that n has ONEprime factor.
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Thanks Bunuel... Very helpful explanation.
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nishantgauragmat
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How many different prime factors does positive integer n have? 18 May 2020, 18:25
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Bunuel
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How many different prime factors does positive integer n have?

(1) 44 < n^2 < 99
(2) 8n^2 has twelve factors


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OFFICIAL SOLUTION:

How many different prime factors does positive integer n have?

(1) 44 < n^2 < 99. This implies that n can be 7, 8, or 9. Each of these numbers have 1 prime: 7, 2, and 3, respectively. Sufficient.

(2) 8n^2 has twelve factors. For \(8n^2=2^3n^2\) to have twelve factors n must be a prime: \(2^3*(prime)^2\) --> number of factors = (3+1)(2+1)=12. Sufficient.

Answer: D.
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Bunuel
Just for clarification,
in Statement 2, n can be any prime number but not 2. As when n = 2, 8n^2 = 32 will have 6 factors. (1,2,4,8,16,32)
Though the answer won't change.
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How many different prime factors does positive integer n have? 13 Jan 2025, 12:36
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