I am trying to show that $7n + 4$ and $5n + 3$ are coprime for all $n \in \mathbb Z$ but I'm stuck.
Please could someone tell me how to show this?
What I tried:
My first attempt was to use that two numbers are coprime iff their $\gcd$ is $1$. So assume that there are integers $k,q$ such that $1 = 7kn + 4k + 5qn + 3q$.
And this is where my first attempt ended as I did not see a way to proceed from there.
My second attempt was to use that if $b> a$ then $\gcd(a,b) = \gcd(b, b-a)$ so
$$ \gcd (7n + 4, 5n + 3) = \gcd (5n + 3, 2n + 1)$$
but this idea also seems to lead nowhere as I don't see how to show that $\gcd (5n + 3, 2n + 1)=1$.