I am given this statement: $a, b \in \mathbb{N}$ are relatively prime if and only if there exist integers $\alpha, \beta$ such that $1 = \alpha \cdot a + \beta \cdot b$.
I know that $\gcd(a^k,b) = 1$ means $a^k$ and $b$ are relatively prime, so if I use this to show $\gcd(a^{k+1}, b) = 1$ then I can prove this by induction.
When $k = 1$, $\gcd(a,b) = 1$ because $a$ and $b$ are relatively prime. So this works for $k = 1$.
Next, I'm assuming the formula holds $\gcd(a^k) = 1$ and now I have to show that $\gcd(a^{k+1}) = 1$.
Because $a^k$ and $b$ are relatively prime, $1 = \alpha \cdot a^k + \beta \cdot b$. I'm pretty sure if I can show that $1 = \alpha \cdot a^{k+1} + \beta \cdot b$, that would complete the proof. This is where I'm stuck.