The standard proof that $\sqrt{2}$ is irrational (for example, the one in Baby Rudin) says that, upon assuming for a contradiction that $\sqrt{2}$ is rational, that we can write $\sqrt{2} = \frac{p}{q}$ where $p$ and $q$ "have no common factors."
This does not seem completely precise to me. What we want to say is that $p$ and $q$ are relatively prime, so that they no common factors other than $1$. Or that they cannot both be even.
Am I correct in this? Is it standard to define relatively prime this way? (In other words, it's so obvious that $1$ is a common factor that we can just discount it?)
this says that if p and q has common factor lets divide both of them by that factor.so at the end we have p and q which doesn't have any common factor.so if we square the equation we can see that p*p=2*q*q, so p is divisible by 2,so if it is divisible by 2 we have that right hand side is divisible by 4.if right hand side is divisible by 4, left is divisible by 4, so q is also divisible by 2,here we have a contradiction that p and q have no prime divisors.