Properties of Modulo Arithmetic

Now that you understand how to add, subtract, and multiply using modulo arithmetic, it's time to look at the properties that help you simplify and understand patterns in modular operations. These properties follow similar rules to normal arithmetic — with a twist!

 

1. Closure Property

Modulo arithmetic is closed under addition, subtraction, and multiplication. This means that:

If $a \bmod n$ and $b \bmod n$ are integers, then:

• $(a + b) \bmod n$
• $(a - b) \bmod n$
• $(a \times b) \bmod n$

... will also give an integer in the range $0$ to $n - 1$.

 

2. Commutative Property

In modulo arithmetic:

• $(a + b) \bmod n = (b + a) \bmod n$
• $(a \times b) \bmod n = (b \times a) \bmod n$

The order of addition or multiplication does not affect the result.

 

3. Associative Property

Grouping does not affect the result of addition or multiplication under modulo.

• $[(a + b) + c] \bmod n = [a + (b + c)] \bmod n$
• $[(a \times b) \times c] \bmod n = [a \times (b \times c)] \bmod n$

 

4. Distributive Property

Multiplication distributes over addition in modulo arithmetic:

$$a \times (b + c) \bmod n = [(a \times b) + (a \times c)] \bmod n$$

This property is especially useful for expanding and simplifying expressions.

 

5. Identity Elements

• $a + 0 \bmod n = a \bmod n$
• $a \times 1 \bmod n = a \bmod n$

0 is the additive identity, and 1 is the multiplicative identity.

 

Worked Example