if a is not equal to b prove that GCD$(a+\sqrt2*b, \sqrt2+1)=1$

Question

I have this question: given that a and b are rational numbers if a is not equal to b: prove that $(a+b\sqrt2)/(\sqrt2+1)$ is irrational. so I proved that $(a+b\sqrt2)$ is irrational. but I need to prove that $GCD(a+b\sqrt2, \sqrt2+1)=1$

Answer 1

So a divisor of $\sqrt{2}+1$ must also divide $b\sqrt{2}+b$. Since a common divisor divides both terms it divides linear combinations of them we know it must divide $b\sqrt{2}+b-(b\sqrt{2}+a)=b-a$, which is a non-zero integer since $a \neq b$, so the divisor must divide this integer. Now by inspecting $\sqrt{2}+1$ we see that the only integers that divide it are $\pm 1$ giving us a greatest common divisor $1$.