For every integer
![d \geq 1](/media/m/2/7/5/275d82dfc679c8016ebdb6bb3c54af92.png)
, let
![M_d](/media/m/a/b/b/abbec76b576f29027b6380c85c8036eb.png)
be the set of all positive integers that cannot be written as a sum of an arithmetic progression with difference
![d](/media/m/f/7/d/f7d3dcc684965febe6006946a72e0cd3.png)
, having at least two terms and consisting of positive integers. Let
![A = M_1](/media/m/b/0/e/b0e4bc33569255ae965466dc5bd7f4d0.png)
,
![B = M_2 \setminus \{2 \}, C = M_3](/media/m/3/c/8/3c84a19f3b7f2018acb9c2feb658393b.png)
. Prove that every
![c \in C](/media/m/0/a/9/0a960ed377ad9f0196b4546d0db08f6b.png)
may be written in a unique way as
![c = ab](/media/m/0/d/a/0da7709828abe33f99c6efaf5f55d33a.png)
with
%V0
For every integer $d \geq 1$, let $M_d$ be the set of all positive integers that cannot be written as a sum of an arithmetic progression with difference $d$, having at least two terms and consisting of positive integers. Let $A = M_1$, $B = M_2 \setminus \{2 \}, C = M_3$. Prove that every $c \in C$ may be written in a unique way as $c = ab$ with $a \in A, b \in B.$