Is it true that numbers of the form $pa+qb = n$ where $\gcd(a,b) = 1$ and $a,b$ are positive integers and $p,q$ are nonnegative integers are unique? That is, they have unique representations $pa+qb$.
I was wondering about this question and wasn't sure. We have that $p_1a+q_1b = p_2a+q_2b$ implying that $a(p_1-p_2) = b(q_2-q_1)$ and thus $k_1b = p_1-p_2$ and $k_2a = q_2-q_1$. We arrive at $k_1 = k_2$.