Let
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
be a given number greater than 2. We consider the set
![V_n](/media/m/b/e/f/beffb5c0823046557932401f8f6def30.png)
of all the integers of the form
![1 + kn](/media/m/1/e/4/1e46bb794e0ef9afb2281f2083816bfd.png)
with
![k = 1, 2, \ldots](/media/m/3/c/a/3ca3ff3912cb1b37d3824ea83937e744.png)
A number
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
from
![V_n](/media/m/b/e/f/beffb5c0823046557932401f8f6def30.png)
is called indecomposable in
![V_n](/media/m/b/e/f/beffb5c0823046557932401f8f6def30.png)
if there are not two numbers
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
and
![q](/media/m/c/1/d/c1db9b1124cc69b01f9a33595637de69.png)
from
![V_n](/media/m/b/e/f/beffb5c0823046557932401f8f6def30.png)
so that
![m = pq.](/media/m/7/b/f/7bf0bee75d528456d558a8f318a3e0d8.png)
Prove that there exist a number
![r \in V_n](/media/m/e/b/9/eb916410a68397756e2c3f8b26c154d6.png)
that can be expressed as the product of elements indecomposable in
![V_n](/media/m/b/e/f/beffb5c0823046557932401f8f6def30.png)
in more than one way. (Expressions which differ only in order of the elements of
![V_n](/media/m/b/e/f/beffb5c0823046557932401f8f6def30.png)
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.)