Amy,

Yes, it is a fact that a number is a square if and only if it has an odd number of divisors. You'd prove this using two things. First, you need the formula for the number of divisors of an integer. This formula is in terms of the exponents in the integer's prime factorization. The second thing that's needed is the fact that an integer is a square if and only if every exponent in its prime factorization is even.

All of these can be found in most elementary number theory books. Good luck!
Victoria