Redaksiya

Firstly, let's assume, that we have only operation choose some $i$, replace $a_i$ by $a_i-k$ (in fact, this operation is enough). Also, let's use it on each element as long, as it is at least $k$. Then, each element will be in range from 0 to $k-1$. After this operations, it is enough to look if all elements are equal. Here is proof:

Let's look at two elements ($x$ and $y$) and their difference ($diff$). When we make some operation on elements ($x$ or $y$), we change value of difference ($diff$) by $k$. So, if they could possibly be equal after some number of operation, then difference should be divisible by $k$. In our array, where each element from 0 to $k-1$, maximum difference is $k-1$, so we can make two elements equal, only if they are already.