The factors of 9 are 1, 3, and 9. Putting 3 on the list a second time is redundant.

Don't let the fact that 3 "pairs" with itself sucker you into thinking that it is somehow different from the other factors. A factor of 9 is any number that divides evenly into 9; it is irrelevant haw many times it does so! In other words, 9/1 = 9 and 9/3 = 3 don't imply that 1 and 3 are somehow different classes of factors because for the first the result is "distinct" and for the second the result is "non-distinct."

It's like this: Andy goes to a certain school in the morning, Billy goes in the morning and afternoon, and Chris goes to school in the afternoon. The students of the school are (Andy, Billy, and Chris) not (Andy, Billy, Billy, and Chris)!

It would be different if the question asked about the "prime factors" of 9.
9 has two prime factors: 3, 3 (Here you do need to repeat, because you need both to "build" 9.)
9 has one distinct prime factor: 3

_________________
Emily Sledge
Instructor
ManhattanGMAT

Can we reverse II and say that
"If the sum of factors of a # is odd, then it is a perfect square"

No. Powers of 2 will meet this condition, whether or not they're squares.

Hi Emily,

Correct me if I am wrong - I have the same doubt as Shikha88 has.

Statement 2 says: The sum of the distinct factors of N is odd.

According to me, if the word distinct would be missing, the statement would not be true

e.g Sum of distinct factors of 9: 1+3+9=13
Sum of factors of 9: 1+3+3+9=16

Hi chitrangada,
Emily is correct. A factor is only counted once, whether or not the problem says "distinct factors". We also don't care whether a factor is multiplied by itself to obtain the original number. So both the factors of 9 and the distinct factors of 9 are listed as 1, 3, 9. The sum of these factors is 13.

Please read Emily's excellent analogy using the counting of schoolchildren as similar to the counting of factors. I hope that after reading her explanation this will make more sense.

Thank you,

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Jamie Nelson
ManhattanGMAT Instructor