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If n is a positive integer, how many different factors n has

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Source: — Data Sufficiency |

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by [email protected] » Wed Aug 23, 2017 7:39 pm
Hi Mo2men,

We're told that N is a positive integer. We're asked for the number of different factors N has. This question can be solved with a mix of TESTing VALUES and Number Property rules.

1) N/5 is a prime number.
IF....
N = 10, the factors are 1, 2, 5 and 10 and the answer to the question is 4
N = 15, the factors are 1, 3, 5 and 15 and the answer to the question is 4
N = 25, the factors are 1, 5 and 25 and the answer to the question is 3
Fact 1 is INSUFFICIENT

2) N has only two different prime factors.
IF....
N = 10, the factors are 1, 2, 5 and 10 and the answer to the question is 4
N = 20, the factors are 1, 2, 4, 5, 10 and 20 and the answer to the question is 6
Fact 2 is INSUFFICIENT

Combined, we know...
N/5 is a PRIME
N has ONLY 2 different prime factors

Since N/5 is prime, one of the two prime factors MUST be a 5 (to 'cancel out' the 5 in the denominator). Whatever the 'other' prime is, it can only show up ONCE in the prime factorization of N. If it showed up more than once, then N/5 would NOT be prime.

For example, 45 = (3)(3)(5), but 45/5 = 9 - which is NOT prime (so N CANNOT be 45).

In addition, the 5 can show up JUST ONCE. If it showed up more than once, then N/5 would NOT be prime.

For example, 75 = (3)(5)(5), but 75/5 = 15 - which is NOT prime (so N CANNOT be 75).

Thus, the 2 different prime factors are the ONLY factors of N that are not 1 or N. Under these conditions, there will ALWAYS be 4 factors:
1
(first prime)
(second prime)
N ... (the product of the two primes).
Combined, SUFFICIENT

Final Answer: C

GMAT assassins aren't born, they're made,
Rich
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If n is a positive integer, how many different factors n has

by Mo2men » Wed Aug 23, 2017 10:35 pm
[email protected] wrote:Hi Mo2men,

We're told that N is a positive integer. We're asked for the number of different factors N has. This question can be solved with a mix of TESTing VALUES and Number Property rules.

1) N/5 is a prime number.
IF....
N = 10, the factors are 1, 2, 5 and 10 and the answer to the question is 4
N = 15, the factors are 1, 3, 5 and 15 and the answer to the question is 4
N = 25, the factors are 1, 5 and 25 and the answer to the question is 3
Fact 1 is INSUFFICIENT

2) N has only two different prime factors.
IF....
N = 10, the factors are 1, 2, 5 and 10 and the answer to the question is 4
N = 20, the factors are 1, 2, 4, 5, 10 and 20 and the answer to the question is 6
Fact 2 is INSUFFICIENT

Combined, we know...
N/5 is a PRIME
N has ONLY 2 different prime factors

Since N/5 is prime, one of the two prime factors MUST be a 5 (to 'cancel out' the 5 in the denominator). Whatever the 'other' prime is, it can only show up ONCE in the prime factorization of N. If it showed up more than once, then N/5 would NOT be prime.

For example, 45 = (3)(3)(5), but 45/5 = 9 - which is NOT prime (so N CANNOT be 45).

In addition, the 5 can show up JUST ONCE. If it showed up more than once, then N/5 would NOT be prime.

For example, 75 = (3)(5)(5), but 75/5 = 15 - which is NOT prime (so N CANNOT be 75).

Thus, the 2 different prime factors are the ONLY factors of N that are not 1 or N. Under these conditions, there will ALWAYS be 4 factors:
1
(first prime)
(second prime)
N ... (the product of the two primes).
Combined, SUFFICIENT

Final Answer: C

GMAT assassins aren't born, they're made,
Rich
Dear Rich,

Thanks a lot for your keen response. However, I have another to see Statement 2 that I understood as follows:
'N has only two different prime factors': For me, it means that I need a number its BASE consisting of TWO Unique prime numbers (Regardless of the exponent of the prime number). For example

10 = 2 * 5................. Number of Factors = 4 (Base prime consists of 2 & 5)
20= 2^2 * 5................Number of Factors = 6 (Base prime consists of 2 & 5)
40= 2^3 * 5...............Number of Factors = 8 (Base prime consists of 2 & 5)
15= 3 * 5...................Number of Factors = 4 (Base prime consists of 3 & 5)
75= 3 * 5^2..............Number of Factors = 6 (Base prime consists of 3 & 5)

Conclusion: Statement 2 is Insufficient

Combining 1 & 2

I reached same conclusion as you did above in your original post.

Does my interpretation wrong for statement 2?

Thanks in advance for your help.
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by [email protected] » Thu Aug 24, 2017 10:00 am
Hi Mo2men,

Your work in Fact 2 is correct. As with any DS question, you have to make sure that you're answering the question that is ASKED (in this case "how many factors does N have?") - and you did. It's worth noting that once you've TESTed 10 and 20, you have enough proof that Fact 2 is insufficient (so you don't have to do any additional work for Fact 2).

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