zuleron wrote:The more interesting question is why do odd numbers behave this way and even numbers do not...

IanStewart wrote:

mendelay wrote:There must be a general rule behind this to avoid plugging numbers. Anyone know?

There is a general rule here, which we can arrive at by extending the logic in maliyeci's excellent explanation above. I could explain this abstractly, but it's probably easier to take a specific example - let's use the number 72 = (2^3)(3^2). Now, this number has 12 factors in total, three of which are odd:

1, 3, 3^2

Now, if we multiply each of the numbers above by 2^1, we get three even divisors of 72, and the same will happen if we multiply these numbers by 2^2 or 2^3. So 72 has three odd divisors, and nine even divisors:

1, 3, 3^2
2, 2*3, 2*3^2
2^2, (2^2)*3, (2^2)(3^2)
2^3, (2^3)*3, (2^3)(3^2)

Notice that we have three times as many even divisors as odd divisors because the power on the 2 in the prime factorization of 72 is 3; that guarantees that we have three even divisors for every odd divisor. You could use this logic for any number, of course, from which we have the following general rule:

* The ratio of the number of even divisors of x to the number of odd divisors of x is always equal to the power on the 2 in the prime factorization of x.

So, if the power on the 2 in the prime factorization of x is equal to 1, we have an equal number of odd and even divisors. If the power is greater than 1, we have more even divisors than odd divisors.