Modulo Arithmetic Operations

Modulo arithmetic, also called clock arithmetic, is all about working with remainders. It’s used to find the remainder when one number is divided by another. This is very common in programming, coding theory, and time calculations (like clocks!).

 

What Does Modulo Mean?

When we say:

$$a \bmod n$$

We mean: what is the remainder when $a$ is divided by $n$?

Example: $17 \bmod 5 = 2$, because when you divide 17 by 5, the remainder is 2.

 

Basic Modulo Arithmetic Operations

1. Addition Modulo $n$

Add the numbers first, then find the remainder when the result is divided by $n$.

$$(a + b) \bmod n = \text{remainder of } a + b \text{ divided by } n$$

Example: $(7 + 9) \bmod 10 = 16 \bmod 10 = 6$

 

2. Subtraction Modulo $n$

Subtract the numbers, then apply the modulo to get a positive result (if needed).

$$(a - b) \bmod n = \text{remainder of } a - b \text{ divided by } n$$

Example: $(4 - 9) \bmod 7 = -5 \bmod 7 = 2$

Because: $-5 + 7 = 2$ (we add 7 to make it positive)

 

3. Multiplication Modulo $n$

Multiply the numbers first, then take the result mod $n$.

$$(a \times b) \bmod n = \text{remainder of } a \times b \text{ divided by } n$$

Example: $(6 \times 7) \bmod 5 = 42 \bmod 5 = 2$

4. Division Modulo (Not Common at WAEC Level)

Division in modulo arithmetic is more advanced and usually not required at the WAEC level. Stick to addition, subtraction, and multiplication for now!

 

Worked Example

 

More Worked Examples

 

Practice Problem