Modular Arithmetic

Grade 7 Math - Modular Arithmetic

Lesson Objectives

Understand the concept of modular arithmetic.
Perform basic calculations using modular arithmetic.
Apply modular arithmetic to solve practical problems.

Lesson Introduction

Have you ever wondered how clocks wrap around after 12 hours or 24 hours? This is an example of modular arithmetic. In this lesson, you'll explore how numbers "wrap around" after reaching a certain value called the modulus.

Core Lesson Content

Definition: Modular arithmetic deals with remainders after division. The notation $a \bmod n$ means "the remainder when a is divided by n".

For example, $14 \bmod 5 = 4$ , because when you divide 14 by 5, the remainder is 4.

Modular arithmetic is used in computer science, encryption, calendars, and clocks.

Worked Examples

Example 1: What is

10 \bmod 3

10 \div 3 = 3 \text{ remainder } 1 \Rightarrow 10 \bmod 3 = 1

Explanation: 3 goes into 10 three times with 1 left over.

Example 2:

25 \bmod 4

25 \div 4 = 6 \text{ remainder } 1 \Rightarrow 25 \bmod 4 = 1

Explanation: 4 times 6 is 24. 25 minus 24 leaves 1.

Example 3:

19 \bmod 5

19 \div 5 = 3 \text{ remainder } 4 \Rightarrow 19 \bmod 5 = 4

Explanation: 5 fits into 19 three times with 4 left.

Example 4: What is

0 \bmod 7

0 \div 7 = 0 \text{ remainder } 0 \Rightarrow 0 \bmod 7 = 0

Explanation: 0 divided by any number gives 0 remainder.

Example 5: If today is Wednesday, what day will it be in 10 days?

10 \bmod 7 = 3 \Rightarrow \text{Wednesday + 3 days = Saturday}

Explanation: Days of the week repeat every 7 days.

Example 6:

100 \bmod 9

100 \div 9 = 11 \text{ remainder } 1 \Rightarrow 100 \bmod 9 = 1

Explanation: 9 times 11 is 99. 100 minus 99 is 1.

Example 7:

47 \bmod 6

47 \div 6 = 7 \text{ remainder } 5 \Rightarrow 47 \bmod 6 = 5

Explanation: 6 fits into 47 seven times with 5 left.

Example 8: Find

18 \bmod 4

18 \div 4 = 4 \text{ remainder } 2 \Rightarrow 18 \bmod 4 = 2

Explanation: 4 times 4 is 16, and 18 - 16 = 2.

Example 9: If it's 9 o'clock now, what time will it be in 10 hours?

(9 + 10) \bmod 12 = 19 \bmod 12 = 7 \Rightarrow \text{7 o'clock}

Explanation: Clock arithmetic is modulo 12.

Example 10: What is

15 \bmod 5

15 \div 5 = 3 \text{ remainder } 0 \Rightarrow 15 \bmod 5 = 0

Explanation: 15 is a multiple of 5, so no remainder.

Exercises

Evaluate $13 \bmod 4$ .
[NABTEC] Find $22 \bmod 6$ . [Past Question]
What is $75 \bmod 10$ ?
If today is Monday, what day will it be in 20 days?
[WAEC] If the current time is 5 o'clock, what will the time be after 17 hours? [Past Question]
[NECO] Find $63 \bmod 8$ . [Past Question]
Evaluate $100 \bmod 11$ .
What is $40 \bmod 3$ ?
[JAMB] Find the remainder when 123 is divided by 12. [Past Question]
What day of the week will it be 50 days from Sunday?

Conclusion/Recap

In this lesson, you learned how modular arithmetic works and how to calculate remainders. You also saw real-life applications such as clocks and calendars. In the next topic, we’ll look at the LCM and HCF.