# Modular Subtraction Rule Proof

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: This site uses Akismet to reduce spam. Learn how your comment data is processed.