# The numbers of the even divisors and the odd divisors of a natural number

Can a natural number have an odd number of the even divisors and an even number of the odd divisors?

Example: All the prime numbers have an even number of the odd divisors (2), but also 0 even divisors (we want odd number of even divisors and 0 is considered as even number)

9 has odd number (3) of odd divisors so it won't fit either

Say, for instance, that our number is divisible by 4 but not 8. Take all the odd divisors, and multiply them by $$2$$. You now have all the even divisors which are not divisible by $$4$$. Multiply them by $$2$$ again, and you have all the divisors which are divisible by $$4$$.
A corresponding argument works no matter how many times $$2$$ goes into our number.