I’ve already presented and proved the rule for modular addition, so for a sense of completeness, but mostly to satisfy my OCD, now I’ll cover the rule for modular subtraction. When doing subtraction in modular arithmetic, the rule is:

If we subtract integer from integer and calculate the difference modulo , we get the same answer as if we had subtracted modulo from modulo and then calculated that difference modulo . Like the modular addition rule, this rule can also be expanded to include multiple integers.

### Proof

In order to prove that the two sides of the equations are equal to one another, we again redefine and using the quotient remainder theorem:

where this means that

where this means that

Starting with the left hand side of the equation, we have:

Eliminating multiples of since we are doing leaves us with:

The right hand side only requires a simple substitution and we’re done: