Divisibility and primes

[srinath.guhan]

Following is the question in cat 2 ( question4) :
If the positive integer N is a perfect square, which of the following must be true?

I. The number of distinct factors of N is odd.
II. The sum of the distinct factors of N is odd.
III. The number of distinct prime factors of N is even.
I only
II only
I and II
I and III
I, II and III

The answer is given as option 3 ( I and II). My question is if we consider a perfect square number, wouldn't the number of distinct factors be even? Why would we not consider 1 here?

[tim]

can you give an example of ANY perfect square where the number of factors is even?

[srinath.guhan]

Well if its sounding silly i cant help it. I need to understand this logic:
usually we don't consider 1 since its a factor of all numbers
But when we are concerned with number of factors that too distinct wouldn't 1 figure here?
Consider 36. Distinct factors would be
1,2,3,4,6,9,12,18

[jnelson0612]

Well if its sounding silly i cant help it. I need to understand this logic:
usually we don't consider 1 since its a factor of all numbers
But when we are concerned with number of factors that too distinct wouldn't 1 figure here?
Consider 36. Distinct factors would be
1,2,3,4,6,9,12,18

You forgot one factor: 36! :-) (for anyone confused, that is an exclamation point, not a factorial)

Let's look at a few examples:
4 has factors 1, 2, 4
9 has factors 1, 3, 9
16 has factors 1, 2, 4, 8, 16

This is the point that Tim is making . . . every perfect square has an odd number of factors. You have pairs of factors (for example, 1 and 16, 2 and 8) and then you have the square root as one distinct factor. The pairs plus the one distinct factor will always give you an odd number of factors.