$a_0|y\implies y-a_0\equiv 0\mod a_0$ $a_0+a_i|y+a_i\implies y-a_0\equiv 0\mod (a_0+a_i)$ where $1\leq i\leq n$ So $y-a_0\equiv 0\mod \text{lcm}(a_0, a_0+a_1, a_0+a_2, \cdots, a_0+a_n)$ So there exists a natural number $m$ s.t. $y=a_0+(m-1)\cdot\text{lcm}(a_0, a_0+a_1, a_0+a_2, \cdots, a_0+a_n)$

What is the relation between $k$ and $n$? To me it's just a typo, where $n$ has replaced $k$ in the second set of relations, so they are the same. Thus the condition is just $\text{lcm}(a_{0}, a_{0}+a_{1}, a_{0}+a_{2},\cdots, a_{0}+a_{k}) \mid y - a_0$, no?

mavropnevma wrote: Thus the condition is just $\text{lcm}(a_{0}, a_{0}+a_{1}, a_{0}+a_{2},\cdots, a_{0}+a_{k}) \mid y - a_0$, no? Yes.