Let
and
be two nonnegative integers. Denote by
the set of numbers
of the form
where
and
are positive integers. Determine
. Prove that if
it is enough to know all the sets
for coprime numbers
in order to know all the sets
. Prove that in the case of coprime numbers
and
contains all numbers greater than or equal to
and also
numbers smaller than
%V0
$(FRA 1)$ 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 $a \neq b,$ 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 $\frac{\omega}{2}$ numbers smaller than $\omega$
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