A little bit of history first. I have asked this question here some days ago. Quoting from there:
An array $A$ is given that has only positive integers in it. The objective is to equalize the array in minimum possible steps. In every step all but one element are incremented by a fixed step size. The possible step sizes are given in another array $D$.For example, if $A=\{2,2,3,7\}$ and $D=\{1,2,5\}$ we can equalize $A$ in two steps, first adding $A=\{5,5,5,0\}$ to get $A=\{7,7,8,7\}$ and then adding $\{1,1,0,1\}$ to get $A=\{8,8,8,8\}$.
If $A$ has just two elements, this reduces to something like a change-making problem. If $\Delta=min(A[0],A[1])$, if I can make change for $\Delta$ using only the given steps from $D=\{1,2,5\}$, I can equalize the array by holding the larger element constant while incrementing the smaller element by the same values that I used to make change for $\Delta$. However,if it's not possible to make change for delta, it's not clear if the two elements can converge two some larger value following this strategy.
I would like to know if this strategy of withholding one element and incrementing the rest by a chosen 'delta' will always make all elements equal in some finite number of steps.