Problem Solving

venmic wrote:I have read in a post and on wiki that 1 need not be considered it can go in everything so not

So if N = 25 then the factors are 5,25 = so the number of factors are even in case of a perfect square

Sum of the factors is even too = 30

PLEASE explain

You certainly need to consider 1 when you count the divisors of a positive integer, so you may have gotten bad information from what you read. Perfect squares have an odd number of divisors. It's easy to see why this should be true. If you take a non-square, like 18, then all of the divisors are in pairs:

1, 18
2, 9
3, 6

If you take a perfect square, however, one divisor will not be in a pair (the square root of the number). For example, the divisors of 36 are

1, 36
2, 18
3, 12
4, 9
6

So perfect squares have an odd number of divisors, and all other numbers have an even number of divisors.

I've solved the rest of the question in the original post a few times elsewhere, but I'd point out that item II in the list (about the sum of a square's divisors) is irrelevant for the GMAT. It's not a property you'd ever want to memorize for the test, since it will never be tested, nor is it the kind of thing you could reasonably be asked to prove is true within two minutes. GMAT questions are never designed so that the only viable strategy is making a guess by picking a few numbers.