OG 11 DS #132

[tonebeeze]

Can someone please walk me through the logic in statement 2.


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.

OA = B

[force5]

hi tonebeeze

stmnt 1 (insuff) eg 2,3,5 etc)
stmnt 2 (sufficient) - 2 ( has only 2 factors 1 and 2 and the difference between 2 and 1 = odd number

3( has factors 1 and 3 the diff will always be even)
4 ( has factors 1,2,4) difference between any 2 distinct factors is not odd.
5 (1 and 5 ) will always be even.

hence you see that the diff between factors is odd only in the case of 2.
hence B

[GMATGuruNY]

Can someone please walk me through the logic in statement 2.


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.

OA = B

Statement 1: n has exactly 2 positive factors.
In other words, n can be any prime number.
Insufficient.

Statement 2: The difference of any two distinct positive factors of n is odd.
Thus, n cannot have two odd factors, since odd-odd = even.
Thus, n cannot have two even factors, since even-even = even.
Thus, n must have exactly one odd factor and exactly one even factor.
Only n=2 has exactly one odd factor (1) and exactly one even factor (2).
Sufficient.

The correct answer is B.

[Stuart@KaplanGMAT]

Can someone please walk me through the logic in statement 2.


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.

Hi,

first off, it's important to understand exactly what (2) says - I've seen many people get this question wrong because they misinterpret (2).

The difference of any two distinct positive factors of n is odd.

This means that if you take any two distinct factors of n, you'll get an odd difference. How is this possible? Only in 1 case: n must have exactly 1 even factor and exactly 1 odd factor.

Let's examine why this conclusion holds:

if n had two odd factors, then if we took the difference between those two factors we'd get an even result and violate the condition;

if n had two even factors, then if we took the difference between those two factors we'd get an even result and violate the condition; and

if n has more than two factors, then we'll always have at least two evens or at least two odds in the mix, leading to a violation of the condition.

Since 2 is the only number that has exactly 1 even factor and exactly 1 odd factor, (2) gives us a definite "yes" answer and is sufficient.