Question

Why does a square number always have an odd amount of factors?

Answer 1

Factors occur in pairs because pairs of factors multiplied together produce the factored number. For example, 12 is produced by multiplying 1 and 12, or 2 and 6, or 3 and 4, so 12 has six different factors. With a perfect square, there is always one factor that produces the number when multiplied by itself. For example, 16 is produced by multiplying 1 and 16, or 2 and 8, or 4 and 4, so there are only five different factors for 16.

Answer 2

let's ask a related question: why do number that aren't squares always have a even number of factors?

this answer is that the factors of a number comes in pairs. for example, the factors of the number 20 are: 1,2,4,5,10, and 20. Because you can write the number 20 as: 1 x 20, 2 x 10, or 4 x 5

for any number that is not a perfect square ever factor will have a partner that you multiply it by in order to get the number. therefore, each factor has a number and therefore the number of factors are always even.

for a perfect square there will always be a lone factor who doesn't have a partner. the question you have to answer is "why is that", to solve your math problem.

hope this helped !!!!!! the best I can do and it's the best way I know to explain it