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If n is a positive integer, does n have four or more distinct factors?

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Bunuel
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If n is a positive integer, does n have four or more distinct factors? [#permalink] New post   02 Mar 2022, 08:30
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If n is a positive integer, does n have four or more distinct factors?

(1) n is not prime

(2) 900 ≤ n < 1100


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PyjamaScientist
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Re: If n is a positive integer, does n have four or more distinct factors? [#permalink] New post   02 Mar 2022, 08:37
(C) in my opinion.

Statement (1): n can be 4 (3 factors: 1,2,4) or n can be 10 (4 factors: 1,2,5,10).
Insufficient alone.

Statement (2): n can be any prime no. between 900 to 1100 and have only two factors, 1 and itself. Or it can be a non-prime and have more than 4 or 4 factors.
Insufficient alone.

Both together, if n is prime and between 900 - 1100, it will only have two factors, 1 and itself. Thus it is sufficient to answer the question. (C)
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BrentGMATPrepNow
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Re: If n is a positive integer, does n have four or more distinct factors? [#permalink] New post   02 Mar 2022, 11:14
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Bunuel wrote:
If n is a positive integer, does n have four or more distinct factors?

(1) n is not prime

(2) 900 ≤ n < 1100

Given: n is a positive integer

Target question: Does n have four or more distinct factors?

Key property: If p is a prime number, then p² will have exactly 3 distinct factors: 1, p and p²
For example, 5 is prime. 5² = 25, and 25 has exactly three distinct factors: 1, 5 and 25

NOTE: In general, if p is prime, and n is a positive integer, then p^n will have exactly n+1 distinct factors.

Given this, we can head straight to....

Statements 1 and 2 combined
Statement 1 tells us that n is not prime
Statement 2 tells us that 900 ≤ n < 1100
There are several values of n that satisfy BOTH statements. Here are two:
Case a: If n = 1000 then the answer to the target question is YES, n has four or more distinct factors
Case b: If n = 31² (note that 31 is prime AND 900 ≤ 31² < 1100) then n has exactly 3 distinct factors (1, 31 and 31²). So, the answer to the target question is NO, n does not have four or more distinct factors
Since we can’t answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E
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Re: If n is a positive integer, does n have four or more distinct factors? [#permalink] New post   02 Mar 2022, 23:37
PyjamaScientist wrote:
(C) in my opinion.

Statement (1): n can be 4 (3 factors: 1,2,4) or n can be 10 (4 factors: 1,2,5,10).
Insufficient alone.

Statement (2): n can be any prime no. between 900 to 1100 and have only two factors, 1 and itself. Or it can be a non-prime and have more than 4 or 4 factors.
Insufficient alone.

Both together, if n is prime and between 900 - 1100, it will only have two factors, 1 and itself. Thus it is sufficient to answer the question. (C)


statement 1 says n is not prime(composite number).
statement 2 provides a range for possible values of n
st. 1 and 2 combined:
31 is a prime number
31*31 = 961 = n
961 is not prime and lies in given range. But it has only 3 factors.
if we pick n = 900, there are more than 4 factors.
Thus, 2 statements combined are not sufficient.

Ans: option E
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Re: If n is a positive integer, does n have four or more distinct factors? [#permalink] New post   18 Mar 2023, 05:51
How do we find out if there are any prime numbers in a range of numbers? Say between 900 and 1100 both inclusive

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Re: If n is a positive integer, does n have four or more distinct factors? [#permalink]
18 Mar 2023, 05:51
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Bunuel
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