Chapter 5 Prime Time

Welcome to the complete tutorial of Chapter 5: Prime Time[cite: 2]. In this lesson, you will explore common multiples, factors, prime and composite numbers, co-prime pairs, prime factorisation, and tests of divisibility through simple explanations, games, and solved examples[cite: 2].

Introduction

Numbers can be grouped and broken down into smaller building blocks[cite: 2]. Understanding how numbers relate to each other through factors and multiples helps us solve mathematical puzzles and real-world problems efficiently[cite: 2].

In this chapter, we delve into Prime Numbers—the fundamental building blocks of all whole numbers[cite: 2].

Did You Know?

The Sieve of Eratosthenes is an ancient algorithm created over 2200 years ago by Greek mathematician Eratosthenes to discover all prime numbers efficiently[cite: 2].

Idli-Vada Game

Understand common multiples of numbers using playful rules[cite: 2].

Jump Jackpot

Discover factors and common factors by taking steps of fixed sizes[cite: 2].

Building Blocks

Prime numbers form every composite number through multiplication[cite: 2].

Safekeeping

Co-prime numbers are essential for cryptography and digital security[cite: 2].

Learning Objectives

After studying this chapter, you will be able to:

Find common multiples and common factors of given numbers[cite: 2].
Distinguish between prime and composite numbers[cite: 2].
Use the Sieve of Eratosthenes to list prime numbers up to 100[cite: 2].
Identify co-prime number pairs[cite: 2].
Express numbers in terms of their prime factorisation[cite: 2].
Apply shortcut divisibility tests for 2, 4, 5, 8, and 10[cite: 2].

Common Multiples and Common Factors

When two or more numbers share the same multiple, it is called a Common Multiple[cite: 2]. When a number divides two numbers exactly without leaving a remainder, it is called a Common Factor[cite: 2].

Factor × Factor = Multiple

Key Concepts

Multiples

Multiples of 3: 3, 6, 9, 12, 15, 18, 21...[cite: 2]
Multiples of 5: 5, 10, 15, 20, 25...[cite: 2]

Common Multiples

First common multiple of 3 and 5 is 15[cite: 2].
Next: 30, 45, 60...[cite: 2]

Factors

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24[cite: 2].
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36[cite: 2].

Common Factors

Common factors of 14 and 36 are 1 and 2[cite: 2].

Prime and Composite Numbers

Numbers can be classified based on how many factors they possess[cite: 2]:

Prime Numbers: Numbers having exactly two factors—1 and the number itself[cite: 2] (e.g., 2, 3, 5, 7, 11, 13, 17, 19...)[cite: 2].
Composite Numbers: Numbers having more than two factors[cite: 2] (e.g., 4, 6, 8, 9, 10, 12...)[cite: 2].
Special Case (1): The number 1 has only 1 factor, so it is neither prime nor composite[cite: 2].

Note: 2 is the smallest prime number and the only even prime number[cite: 2]. All other even numbers are composite[cite: 2].

Co-prime Numbers

Two numbers are said to be co-prime if they have no common factor other than 1[cite: 2].

Number Pair	Common Factors	Status
4 and 9	1	Co-prime[cite: 2]
15 and 39	1, 3	Not Co-prime[cite: 2]
18 and 35	1	Co-prime[cite: 2]

Prime Factorisation

Expressing a composite number as a product of only prime numbers is called Prime Factorisation[cite: 2]. Every whole number greater than 1 has a unique prime factorisation (except for the order of factors)[cite: 2].

Example: Prime Factorisation of 56

56 = 4 × 14 = (2 × 2) × (2 × 7) = 2 × 2 × 2 × 7[cite: 2]

Using Prime Factorisation to Check Co-Primes:

If two numbers share no common prime factors, they are co-prime[cite: 2]!

Divisibility Tests

Divisibility rules allow us to quickly determine if a large number is divisible by a smaller number without doing long division[cite: 2].

Divisibility By	Rule	Example
2	Ends in 0, 2, 4, 6, or 8[cite: 2]	682, 8560[cite: 2]
4	Last two digits are divisible by 4[cite: 2]	8536 (36 ÷ 4 = 9)[cite: 2]
5	Ends in 0 or 5[cite: 2]	2345, 8560[cite: 2]
8	Last three digits are divisible by 8[cite: 2]	8560 (560 ÷ 8 = 70)[cite: 2]
10	Ends in 0[cite: 2]	8560, 980[cite: 2]

Solved Examples

Example 1

Find the common factors of 20 and 28[cite: 2].

Factors of 20 = 1, 2, 4, 5, 10, 20[cite: 2]

Factors of 28 = 1, 2, 4, 7, 14, 28[cite: 2]

Common Factors = 1, 2, 4[cite: 2]

Example 2

Find the prime factorisation of 108 × 75[cite: 2].

108 = 2 × 2 × 3 × 3 × 3[cite: 2]

75 = 3 × 5 × 5[cite: 2]

Answer = 2 × 2 × 3 × 3 × 3 × 3 × 5 × 5[cite: 2]

Example 3

Check if 40 and 231 are co-prime[cite: 2].

Prime factors of 40: 2, 5[cite: 2]

Prime factors of 231: 3, 7, 11[cite: 2]

No common prime factors → Co-prime![cite: 2]

Practice Yourself

Find all multiples of 40 that lie between 310 and 410[cite: 2].
Find three pairs of prime numbers less than 20 whose sum is a multiple of 5[cite: 2].
Are 57 and 85 co-prime?[cite: 2]
Find the prime factorisation of 1000[cite: 2].
Is 14560 divisible by 8 and 5?[cite: 2]

Chapter Summary

If a number divides another completely, it is a factor of the second number[cite: 2].
Prime numbers have exactly 2 factors (1 and itself)[cite: 2].
Composite numbers have more than 2 factors[cite: 2].
1 is neither prime nor composite[cite: 2].
Numbers with no common factor other than 1 are co-prime[cite: 2].
Prime factorisation represents a composite number as a product of primes[cite: 2].
Divisibility rules make quick mental calculations easy[cite: 2].

Multiple Choice Questions

1. Which of the following is the only even prime number?[cite: 2]

A. 0
B. 1
C. 2
D. 4

Answer : C[cite: 2]

2. What is the first number for which 'idli-vada' (common multiple of 3 and 5) is said?[cite: 2]

A. 5
B. 10
C. 15
D. 30

Answer : C[cite: 2]

3. Which of these pairs is co-prime?[cite: 2]

A. 15 and 39
B. 4 and 9
C. 18 and 81
D. 20 and 55

Answer : B[cite: 2]

4. A number ending with 0 is divisible by[cite: 2]:

A. 2, 5, and 10
B. 3 only
C. 7 only
D. 9 only

Answer : A[cite: 2]

5. How many prime numbers exist between 1 and 10?[cite: 2]

A. 2
B. 3
C. 4
D. 5

Answer : C (2, 3, 5, 7)[cite: 2]

Practice Questions

Define prime and composite numbers with two examples each[cite: 2].
Find seven consecutive composite numbers between 1 and 100[cite: 2].
Explain twin primes and give three examples below 50[cite: 2].
Write two numbers whose product is 10000, neither having 0 as unit digit[cite: 2].
State the divisibility rule for 4 and verify for 8536[cite: 2].