Question

I am trying to find the possible values of this expression.

(a mod 2*x)-(a mod x)

I suspect they might be 0 or x, but I’m really not sure. I can’t seem to be able to write down a proper argument.

Answer 1

You are correct that the possible values are 0 and x, assuming that both a and x are positive. The logic is as follows.

Let a have the form

a = p*x + b

Then it is easy to see that a mod x = b.

For a mod 2*x, if p = 2*r (p is even) then

a = 2*r*x + b = (2*x)*r + b

so that a mod 2*x = b and p = 2*r + 1 (p is odd) then

a = (2*r + 1)*x + b = 2*r*x + x + b = (2*x)*r + x + b

so that a mod 2*x = x + b. Combining these results, the difference is either b - b = 0 (when p is even) or (x + b) - b = x (when p is odd).