![(FRA 1)](/media/m/9/e/8/9e88828b15db8417ec3f3e6b73bd930c.png)
Let
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
and
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
be two nonnegative integers. Denote by
![H(a, b)](/media/m/5/3/6/53623c9ebe9b04f9e6a8337f85d1c585.png)
the set of numbers
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
of the form
![n = pa + qb,](/media/m/5/a/2/5a22f19d3e3ecf714a318f1e9d25d578.png)
where
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
and
![q](/media/m/c/1/d/c1db9b1124cc69b01f9a33595637de69.png)
are positive integers. Determine
![H(a) = H(a, a)](/media/m/7/6/8/768d4e10a194b2e6e57aaab6e1284faa.png)
. Prove that if
![a \neq b,](/media/m/7/e/c/7ecb3fcb5acd7e76b44db3005a32d640.png)
it is enough to know all the sets
![H(a, b)](/media/m/5/3/6/53623c9ebe9b04f9e6a8337f85d1c585.png)
for coprime numbers
![a, b](/media/m/a/2/b/a2bdbf048e2daac0a021a1d79f6fb9bf.png)
in order to know all the sets
![H(a, b)](/media/m/5/3/6/53623c9ebe9b04f9e6a8337f85d1c585.png)
. Prove that in the case of coprime numbers
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
and
![b, H(a, b)](/media/m/4/9/e/49e93b02bc99c9dbddeaaea477f5e41f.png)
contains all numbers greater than or equal to
![\omega = (a - 1)(b -1)](/media/m/c/d/c/cdcdb254f16ac5e6abb3200fd8a1d314.png)
and also
![\frac{\omega}{2}](/media/m/6/4/b/64ba34e1f559f10e67e7bdf1eda3751c.png)
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 $\omega$