Explain why all square numbers have an odd number of factors?

Answer 1

Let's take an example. 12 has the prime factors 22 x 3. If you square that, you'll get 24 x 32. Note that all the exponents in the square are even. The number of all possible factors (in this example) is (4+1)(2+1), since you can take each factor any number of times between zero, and the exponent for that factor - for example, you can use the factor 2 any number of times between zero and 4. Because of the added 1, all the factors are odd - and if you multiply only odd numbers, the result will also be odd.