As I was riding home from work today I was thinking about odd and even numbers.

It is a funny thing that the product of two even numbers must be an even number, while the product of an even number with an odd number must also be even. Only two odd numbers will always give an odd number when multiplied.

If you don't believe me, think about what makes the number odd or even, it is whether there is a remainder of one after you divide by two. When you multiply an odd by an odd, it is the same as multiplying the first odd number by the second number minus one, and then adding the first number. The first operation must give you an even number (odd times even) so that then adding an odd number must give you an odd.

This tells us some interesting things, firstly only even numbers can have factors that are both odd and even. Odd numbers will only ever have odd factors.

It also means that if you take two random numbers then the probability of the product being odd is just 1/4. The reason is that there are 4 possible ways to draw two random numbers: odd+odd, odd+even, even+odd, even+even. Only one of those 4 options can produce an odd number.

This result could also mean that in general even numbers have more factors than odd numbers. I don't have an argument for it, but it seems to me to be the kind of thing for which there might be a formal proof, perhaps I was even shown it and have forgotten. If you know of one please point it out in the comments.

Anyway, these thoughts passed the time as I rode home today and helped me clear my mind of other things. Who would have thought that amateur number theory could be so satisfying.