Given k % y, how can I adjust the dividend (k) to preserve the modulo when the divisor (y) is incremented by one?

Question

In a programming algorithm, I'm using the result of k % y. I need to understand how to adjust k when the value of y is incremented by one to preserve the same modulo result.

In other words, solve for x: (k + x) % (y + 1) = k % y

Empirically, I see that I can always adjust k by adding a constant that depends on both the value of k and the value of y.

I believe the solution involves computing some sort of "distance across y" between k % y and k % (y + 1).

Hoping this is easy for someone who studied mathematics, rather than comp sci, as I'm out of my league here. Thanks!

Answer 1

You have $k=my+r$ for some $m,r$ and $k\%y=r=k-my$. Now if you want $k'\%(y+1)=r$, you need $k'=m'(y+1)+r$. One way to assure this is to have $m'=m, k'=k+m$. Unfortunately, just having $k\%y$ you don't know $m$. Another way to assure this is to let $k=r$ but it seems you want $k'$ to depend upon $k$ somehow. Yet a third way is to set $k'=k(y+1)+r$ Do any of these meet your needs?