Why do odd numbers have more factors than even ones?

Answer 1

They do?

There are 6 factors of 32 an even number (1, 2, 4, 8, 16, 32), but only 4 factors of 33 an odd number (1, 3, 11, 33).

This one example pair of an odd number and an even number contradicts your assertion that odd numbers have more factors than even numbers.

The number of factors of a number depends upon the prime factorisation of the number - even or oddness of the number has no bearing on the number of factors of the number.

If one of the primes (in the factorisation of a number) happens to be 2 (the only even prime), then the number will be even.

If a number's prime factorisation is represented in power format (eg 44 = 22 x 11), then the total number of factors of the number is given by the product of the powers of the primes incremented by one, that is:

number = Π pini

where

pi are the primes

ni are the corresponding powers of the primes

then

number_of_factors = Π (ni + 1)

examples:

• 44 = 22 x 11

⇒ number_of_factors = (2 + 1) x (1 + 1) = 3 x 2 = 6

The factors of 44 are: 1, 2, 4, 11, 22, 44

• 1500 = 22 x 3 x 53

⇒ number_of_factors = (2 + 1) x (1 + 1) x (3 + 1) = 3 x 2 x 4 = 24

The factors of 1500 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 125, 150, 250, 300, 375, 500, 750, 1500