Data Sufficiency

If the integer n is greater than 1, is n equal to 2?
(1) n has exactly two positive factors
(2) The difference between any two distinct positive factors is odd.

Please provide answer with explanation.

If n > 1, is n = 2?

Statement (1): n has exactly two positive factors. So, 'n' can be any prime number. n = 2, 3, 5.. and so on
INSUFFICIENT

Statement (2): The wording of the statement is not clear. If the statement means to say that 'The difference between the only two positive factors of n is an odd integer', then this statement is SUFFICIENT.

This seems like a problem made by a well-meaning amateur.

amp0201 wrote:If the integer n is greater than 1, is n equal to 2?
(1) n has exactly two positive factors
(2) The difference between any two distinct positive factors is odd.

Hi!

There's nothing ambiguous about (2), it just needs to be read carefully (like all DS statements!).

"The difference between ANY two distinct positive factors of n is odd" must mean two things:

1) n only has 2 distinct factors; and
2) those 2 factors are 1 apart.

The first criterion tells us that n is prime; the second criterion tells us that n=2.

Since the only value that satisfies both of those criteria is 2, statement (2) provides a definite YES answer to the question and is sufficient.

Stuart,

I understand what you are saying. But I assumed n = 6 (distinct factors - 1,2,3,6) and difference between any two distinct factors (6-1) = ODD. Hence n = 2, 6, etc.

So as per your explanation does that mean (2) is looking for ONLY prime numbers, whose factors differ by 1 i.e. difference is odd?

Regards,
Akhil

Aneesh - Thanks for your feedback.

amp0201 wrote:Stuart,

I understand what you are saying. But I assumed n = 6 (distinct factors - 1,2,3,6) and difference between any two distinct factors (6-1) = ODD. Hence n = 2, 6, etc.

So as per your explanation does that mean (2) is looking for ONLY prime numbers, whose factors differ by 1 i.e. difference is odd?

Regards,
Akhil

Aneesh - Thanks for your feedback.

I suppose you forgot to consider 6 - 2 = even

amp0201 wrote:Stuart,

I understand what you are saying. But I assumed n = 6 (distinct factors - 1,2,3,6) and difference between any two distinct factors (6-1) = ODD. Hence n = 2, 6, etc.

So as per your explanation does that mean (2) is looking for ONLY prime numbers, whose factors differ by 1 i.e. difference is odd?

Regards,
Akhil

Aneesh - Thanks for your feedback.

Hi! As Sanju points out, 6-2 is, in fact even; as is 3-1, another pair of factors of 6.

Let's think about statement (2) some more. How do we get an odd difference between integers? If one is even and one is odd.

So, if a number has two odd factors, then we'll get an even difference. If a number has two even factors, we'll get an even difference. The only way to guarantee that we'll ALWAYS get an odd difference is if the number has exactly 1 even factor and 1 odd factor - and only 2 fits that bill.

There is ambiguity in the second statement.

ronnie1985 wrote:There is ambiguity in the second statement.

Hi Ronnie,

what, exactly, is ambiguous? I'd argue that there's only one way that it can be properly interpreted (the word "any" is the key to proper interpretation).

I think answer should be (C)
2) says difference is odd but not 1 apart? what if n=12 (1,2,3,4,6,12). 6-3 is odd.

Not sure if I am missing something but C seems to be the right answer.

PGMAT wrote:I think answer should be (C)
2) says difference is odd but not 1 apart? what if n=12 (1,2,3,4,6,12). 6-3 is odd.

Not sure if I am missing something but C seems to be the right answer.

Hi,

the key word is "any".

(2) says that the difference between ANY two factors of n is odd; read ANY as EVERY (they mean the same thing).

So, if n=12, then we have lots of pairs of factors that do NOT have an odd difference, e.g.:

3-1=2
4-2=2
6-2=4

and so on...

The only number for which ANY two factors chosen have an odd difference is 2.