I am slightly stumped by this answer explanation given in OG11 (DS, Q132, Pg 331). Would appreciate it if someone could explain, thanks!
Q: If the integer n is greater than 1, is n equal 2?
1) n has exactly two positive factors
2) The difference of any two distinct positive factors of n is odd.
[spoiler Answer=B, statement 2 is sufficient]
OG's explanation:
1) If n has exactly two positive factors, then n is a prime number. However, n could be any prime number so statement 1 is not sufficient. (I agree)
2) [Copying their explanation word for word here.] "Note that if n>2 and n is odd, then 1 and n are factors of n, and their difference is even. Also, if n >2 and n is even,then 2 and n are factors of n, and their difference is even. Thus, no integer greater than 2 satisfies this statement. However, n=2 does satisfy this statement since 1 and 2 are the only positive factors of 2, and their difference is odd; Statement 2 alone is sufficient."
My question: Statement 2 only states that the difference of "ANY two distinct positive factors" of n is odd. I literally took this to mean ANY and not ONLY..I.e. 3 and 2 are distinct factors of 6 and their difference (=1) is odd. Therefore n can be 2 or n can be 6 and we cannot establish a unique value for n using statement 2.
However, using statements 1 and 2 combine, we can eliminate all non-prime numbers and the only prime number where its two distinct positive factors are odd is 2. Therefore, statements 1 and 2 are sufficient, in my opinion, but not statement 2 alone.
Where have I gone wrong? [/spoiler]
Q: If the integer n is greater than 1, is n equal 2?
1) n has exactly two positive factors
2) The difference of any two distinct positive factors of n is odd.
[spoiler Answer=B, statement 2 is sufficient]
OG's explanation:
1) If n has exactly two positive factors, then n is a prime number. However, n could be any prime number so statement 1 is not sufficient. (I agree)
2) [Copying their explanation word for word here.] "Note that if n>2 and n is odd, then 1 and n are factors of n, and their difference is even. Also, if n >2 and n is even,then 2 and n are factors of n, and their difference is even. Thus, no integer greater than 2 satisfies this statement. However, n=2 does satisfy this statement since 1 and 2 are the only positive factors of 2, and their difference is odd; Statement 2 alone is sufficient."
My question: Statement 2 only states that the difference of "ANY two distinct positive factors" of n is odd. I literally took this to mean ANY and not ONLY..I.e. 3 and 2 are distinct factors of 6 and their difference (=1) is odd. Therefore n can be 2 or n can be 6 and we cannot establish a unique value for n using statement 2.
However, using statements 1 and 2 combine, we can eliminate all non-prime numbers and the only prime number where its two distinct positive factors are odd is 2. Therefore, statements 1 and 2 are sufficient, in my opinion, but not statement 2 alone.
Where have I gone wrong? [/spoiler]
