Show that $\sqrt{10}$ is irrational using the Fundamental theorem of arithmetic. I know I can prove it by way of contradiction. But I also want to know how to do this with this method.
By contradiction, $\sqrt{10}$ is rational. Then we can write $\frac{a}{b} = \sqrt{10}$ where a and b are integers and b is not 0 and the gcd(a,b) = 1. Then $10 = \frac{a^2}{b^2} => 10b^2 = a^2$. Because $a^2$ is even, a is even, and then we can write a = 2k where k is an integer. Then $10b^2 = (2k)^2 => 10b^2 = 4k^2 => 10b^2 = 2(2k^2) => 5b^2 = 2k^2$
We do not have a gcd(a,b) = 1 Because our assumption was wrong, $\sqrt{10}$ is irrational.
I want to know how to use the theorem. I looked at it and I got confused. Thanks