factors

[mmesarina]

From Gmat review 11th

If the integer n is greater than 1 , is n equal to 2?

1) n has exactly two positive factors
2) the difference of any two distinct positive factors of n is odd.

SPOILER: B

Can someone tell me what am I doing wrong here?

From stmt1) n could be any number so Insufficient

from stmt 2) say n has two factors a and b , so that n = a * b
If a-b or b-a is odd, that means that either a=even, b=odd, OR a=odd, b=even, in any there are many numbers that fit this description for a and b.
Now, the question statement asks to find whether n = 2.
If n =2, then there can only be two factors, such that 2 = 1 * 2. So the difference of 1 and 2 is always odd, which fits the description of stm2, but there are many combinations of numbers that fit stm2 description. So my answer was E.

Could someone please explain the Official Answer?

[avikar]

For stmnt 2, I took 4 = 2*2*1 the difference two distinct positive factors of n is odd (2-1).

Stmnt 2 doesnot say that n has only two factors, right?

Can some please explain my mistake? Thanks.

[tank]

1 and the number(n) itself are factors of the number.

Now, if the number is odd, n-1 must be even. So statement2 does not hold.

Now, for n being even, consider 2 cases: n>2 and n=2
For n>2 and n being even, 2 has to be a factor of n. However, since n is even, n-2 is even. So statement2 does not hold.

Now for n=2, the only factors are 1 and 2 and the difference is odd.

Hence statement2 is enough.

[avikar]

1 and the number(n) itself are factors of the number.

Now, if the number is odd, n-1 must be even. So statement2 does not hold.

Now, for n being even, consider 2 cases: n>2 and n=2
For n>2 and n being even, 2 has to be a factor of n. However, since n is even, n-2 is even. So statement2 does not hold.

Now for n=2, the only factors are 1 and 2 and the difference is odd.

Hence statement2 is enough.

Thanks Tank!! now i see my mistake. For 4 --> 1, 2 and 4 are factors, before I was considering 1, 2 (only prime factors)