Numbers on steroids

AS some of you are aware, I am trying my hand at the GMAT. Nothing serious. I was just bored over a couple of weekends. I Didn't have any code to write so I decided that I was going to write the GMAT. These are three properties regarding perfect squares which I learnt about and I am putting them here for any person who is writing the GMAT. All right I am putting it down here so that I can google myself up later.

These are the props :

1. A perfect square has an even number of powers of prime factors
2. Any perfect square integer always has an odd number of distinct factors.
3. For any perfect square, the sum of its distinct factors is always odd

Now for the substantiations: 

• A perfect square has an even number of powers of prime factors

This is "THE BASIC RULE" and the other two (and perhaps many others) rules can be derived out of this rule. The rule says - For a perfect square, N, if N is prime factorized, say N = (p^x) * (q^y), x and y can and will only be even integers. This seems pretty obvious. If x and/or y were infact odd, there wouldn't be able to find  sqrt(N) in an even positive integer.

With  that now settled, in order to prove the second point,

• Any perfect square integer always has an odd number of distinct factors.

there is this tiny hack that lets you find the number of distinct factors a number has. it is a simple two step process. 

1. Factor the number into its prime components, N = (p^x) * (q^y)
2. The number of distinct factors =(x+1)*(y+1).

Since we proved just now that for such a perfect square x and y will be even,
 the number of factors = (even + 1)*(even + 1)
                                    = odd* odd
                                    = odd

 The third rule is a bit tricky

• For any perfect square, the sum of its distinct factors is always odd

I havent arrived at a thorough proof for this one as of now. But one thing to remember is that any perfect square will have an odd number of odd factors and an even number of even factors. So adding all these up, we get an odd integer. Try it.