Let
be a given number greater than 2. We consider the set
of all the integers of the form
with
A number
from
is called indecomposable in
if there are not two numbers
and
from
so that
Prove that there exist a number
that can be expressed as the product of elements indecomposable in
in more than one way. (Expressions which differ only in order of the elements of
will be considered the same.)
%V0
Let $n$ be a given number greater than 2. We consider the set $V_n$ of all the integers of the form $1 + kn$ with $k = 1, 2, \ldots$ A number $m$ from $V_n$ is called indecomposable in $V_n$ if there are not two numbers $p$ and $q$ from $V_n$ so that $m = pq.$ Prove that there exist a number $r \in V_n$ that can be expressed as the product of elements indecomposable in $V_n$ in more than one way. (Expressions which differ only in order of the elements of $V_n$ will be considered the same.)