Question

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.

Answer 1

I. The number of distinct factors of N is odd

Explanation:

Here, we assume that "factors" means "divisors", rather than "prime factors."

When divisor d is paired with N/d, all divisors less than √N will be paired with divisors greater than √N. Taken together, those constitute an even number of divisors. The divisor √N will remain unpaired (except with itself), so there will be a total of an odd number of divisors.

Simple example:

4 is a perfect square. Its divisors are 1, 2, 4. The pair 1*4 matches the one divisor less than √4 with its counterpart greater than √4. That leaves divisor √4 = 2 with no counterpart except itself.

6 is not a perfect square. Its divisors are 1, 2, 3, 6. When paired, there are none left over: 6 = 1*6 = 2*3.