On Coprime Numbers and Age Differences Hypothetical Situation: Currently, I am $38$ and my wife is $33$, so this year our ages are coprime. Will our ages always be coprime? (If we happen to have the same birthday, so our age difference remains constant).
In general, if any two numbers $n$ and $n+\alpha$ are coprime, will they be coprime for all $n$?
I can see that this is true when $\alpha =1$, obviously. I can also see that $\alpha$ can never be even, because if $n$ and $\alpha$ are even, $n + \alpha$ would also be even, and therefore not coprime.
In this hypothetical situation, $n = 33$ and $\alpha = 5$. I see that my original hypothesis is incorrect, because if $n=10$, then $n+\alpha =15$, and therefore not coprime.
So my conjecture is: $\forall \{n, \alpha \} \in \Bbb{N}$, $n$ and $n+\alpha$ are always coprime iff $n$ and $\alpha$ are coprime. Is this true, or are there more general cases or restrictions?

Yes, $n$ and $\alpha$ are coprime  $\iff n$ and $n+\alpha$ are coprime.
A property of $\gcd$ is that $\gcd(a,b) = \gcd(a+kb, b)$ for any $k\in\mathbb{Z}$. The proof is very easy.
