Question 18
(FRA 1) Let $a$ and $b$ be two nonnegative integers. Denote by $H(a,b)$ the set of numbers $n$ of the form $n=pa+qb$, where $p$ and $q$ are positive integers. Determine $H(a)=H(a,a)$. Prove that if $a\ne b$, it is enough to know all the sets $H(a,b)$ for coprime numbers $a,b$ in order to know all the sets $H(a,b)$. Prove that in the case of coprime numbers $a$ and $b$, $H(a,b)$ contains all numbers greater than or equal to $\omega=(a-1)(b-1)$ and also $\omega/2$ numbers smaller than $\omega$.