The Prime Bet

Let's say you sit in a pub, minding your own business, when all of a sudden a stranger walks up to you and offers you a bet:

We'll choose two positive integers at random. If they have any divisor in common (other than $1$) I'll pay you a dollar, else you'll pay me a dollar. Are you in?

Apart from the question what kind of establishments you frequent, you should be wondering: is this a good bet for you?

When two integers have no divisors in common except the trivial divisor $1$ we say they are coprime or relatively prime. $6$ and $9$ have the common divisor $3$, so they are not coprime, whilst $8$ and $9$ only have the trivial common divisor $1$, so they are coprime.

This makes you start thinking: "As numbers grow bigger, aren't there a lot of divisors out there? After all, half the numbers are even, so if we hit two even numbers, they'll have the factor $2$ in common and I'll win. And then there's $3$, $5$, $7$, ... Seems like a good deal!" Continue reading The Prime Bet