Prime Time class 6th notes
Class 6 Mathematics
Chapter 5: Prime Time
Discover factors, multiples and the fascinating world of prime numbers.
1. Learning Objectives
By the end of this lesson, you should be able to:
- Understand factors and multiples.
- Distinguish between prime and composite numbers.
- Recognise that 1 is neither prime nor composite.
- Use divisibility rules to test numbers.
- Find prime factors using factor trees.
- Understand common factors and common multiples.
- Use the Sieve of Eratosthenes to identify prime numbers.
2. Factors of a Number
A factor of a number divides it exactly, leaving no remainder.
Example: Factors of 12
We can arrange 12 objects into equal groups in these ways:
One row of 12 objects, or 12 rows of one object.
Two rows of six objects, or six rows of two objects.
Three rows of four objects, or four rows of three objects.
The factors of 12 are:
Factors of 12 = 1, 2, 3, 4, 6, 12How to Find Factors
- Start with 1 and the number itself.
- Test other whole numbers to see whether they divide exactly.
- List every number that divides without a remainder.
Worked Example: Factors of 18
| Multiplication pair | Factors found |
|---|---|
| 1 × 18 = 18 | 1, 18 |
| 2 × 9 = 18 | 2, 9 |
| 3 × 6 = 18 | 3, 6 |
3. Multiples of a Number
A multiple is the result of multiplying a number by a whole number.
Example: Multiples of 4
4 × 1 = 44 × 2 = 8
4 × 3 = 12
4 × 4 = 16
4 × 5 = 20
Therefore, the first five positive multiples of 4 are 4, 8, 12, 16 and 20.
Factors vs Multiples
| Factors | Multiples |
|---|---|
| Factors of 6: 1, 2, 3, 6 | Positive multiples of 6: 6, 12, 18, 24, ... |
| A factor divides the number exactly. | A multiple is obtained by multiplying the number. |
| A positive number has a finite number of positive factors. | A positive number has infinitely many positive multiples. |
Common Factors
Find the factors of 12 and 18.
Factors of 12: 1, 2, 3, 4, 6, 12Factors of 18: 1, 2, 3, 6, 9, 18
The numbers appearing in both lists are common factors.
Common factors = 1, 2, 3, 6Common Multiples
Consider the positive multiples of 3 and 4.
Multiples of 3: 3, 6, 9, 12, 15, 18, 24, ...Multiples of 4: 4, 8, 12, 16, 20, 24, ...
Some common multiples are 12, 24, 36 and so on.
4. Prime and Composite Numbers
Prime Number
A prime number is a whole number greater than 1 that has exactly two positive factors: 1 and itself.
Examples: 2, 3, 5, 7, 11, 13
Composite Number
A composite number is a whole number greater than 1 that has more than two positive factors.
Examples: 4, 6, 8, 9, 10, 12
Why Is 7 Prime?
Factors of 7 = 1, 7It has exactly two positive factors, so 7 is prime.
Why Is 12 Composite?
Factors of 12 = 1, 2, 3, 4, 6, 12It has more than two positive factors, so 12 is composite.
The number 1 has only one positive factor: itself. Therefore, 1 is neither prime nor composite.
Two is the smallest prime number and the only even prime number. Every other even whole number greater than 2 is composite because it is divisible by 1, 2 and itself.
Prime Numbers from 1 to 50
The following grid highlights prime numbers in yellow and composite numbers in light blue. The number 1 is marked separately.
Prime numbers from 1 to 50:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 475. The Sieve of Eratosthenes
The Sieve of Eratosthenes is a method for finding prime numbers by crossing out composite numbers.
Interactive Prime Number Activity
Click Start Sieve to identify prime numbers from 1 to 100.
Steps of the Sieve
- Write the numbers from 1 to 100.
- Mark 1 separately because it is not prime.
- Start with 2. Keep 2 and cross out its larger multiples.
- Move to the next number that has not been crossed out.
- Keep that number and cross out its larger multiples.
- Continue until the process is complete.
- The remaining numbers greater than 1 are prime.
6. Divisibility Rules
Divisibility rules help us check whether one number divides another exactly without performing full division.
| Divisor | Rule | Example |
|---|---|---|
| 2 | The last digit is 0, 2, 4, 6 or 8. | 246 is divisible by 2. |
| 3 | The sum of the digits is divisible by 3. | 123: 1 + 2 + 3 = 6. |
| 4 | The number formed by the last two digits is divisible by 4. | 316: 16 is divisible by 4. |
| 5 | The last digit is 0 or 5. | 375 is divisible by 5. |
| 6 | The number is divisible by both 2 and 3. | 132 is divisible by 2 and 3. |
| 9 | The sum of the digits is divisible by 9. | 729: 7 + 2 + 9 = 18. |
| 10 | The last digit is 0. | 540 is divisible by 10. |
Worked Examples
Example 1: Is 234 divisible by 3?
2 + 3 + 4 = 9Since 9 is divisible by 3, 234 is divisible by 3.
Example 2: Is 428 divisible by 4?
The last two digits are 28. Since 28 ÷ 4 = 7, the number 428 is divisible by 4.
Example 3: Is 246 divisible by 6?
The number is even, so it is divisible by 2. Also, 2 + 4 + 6 = 12, which is divisible by 3. Therefore, 246 is divisible by 6.
7. Prime Factorisation
Prime factorisation means expressing a composite number as a product of prime numbers.
Example: Prime Factorisation of 24
We can split 24 into smaller factors until all the final factors are prime.
All the final numbers are prime.
24 = 2 × 2 × 2 × 3Another way to write this is:
24 = 2³ × 3Here, 2³ means 2 × 2 × 2.
Another Example: Prime Factorisation of 36
36 = 4 × 936 = 2 × 2 × 3 × 3
36 = 2² × 3²
8. Practice Worksheet
Part A: Basic Questions
- Write all the factors of 16.
- Write the first six positive multiples of 7.
- Is 19 prime or composite? Give a reason.
- Why is 1 neither prime nor composite?
- What is the only even prime number?
Part B: Divisibility
- Check whether 456 is divisible by 2, 3 and 4.
- Is 735 divisible by 5 and 9?
- Check whether 624 is divisible by 6.
- Is 1,230 divisible by 10?
Part C: Challenge Questions
- Find the common factors of 20 and 30.
- Write the common multiples of 4 and 6 that are less than 50.
- Find the prime factorisation of 40.
- Find the prime factorisation of 60.
- A teacher has 24 pencils. Can they be divided equally among 3, 4, 5 or 6 students without leftovers? Explain.
Show worksheet answers
Part A
- 1, 2, 4, 8, 16.
- 7, 14, 21, 28, 35, 42.
- 19 is prime because its only positive factors are 1 and 19.
- 1 has only one positive factor, not exactly two or more than two.
- 2.
Part B
- 456 is divisible by 2 because it ends in 6. It is divisible by 3 because 4 + 5 + 6 = 15. It is divisible by 4 because 56 is divisible by 4.
- 735 is divisible by 5 because it ends in 5. It is not divisible by 9 because 7 + 3 + 5 = 15, which is not divisible by 9.
- 624 is even, and 6 + 2 + 4 = 12 is divisible by 3. Therefore, 624 is divisible by 6.
- Yes. It ends in zero.
Part C
- Common factors: 1, 2, 5, 10.
- 12, 24, 36, 48.
- 40 = 2 × 2 × 2 × 5 = 2³ × 5.
- 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.
- Yes for 3, 4 and 6 because 24 is divisible by each. No for 5 because 24 is not divisible by 5.
9. Interactive Quiz
Choose one answer for each question and click Check Answers.
10. Quick Revision
- Factor: divides a number exactly.
- Multiple: obtained by multiplying a number by a whole number.
- Prime number: has exactly two positive factors.
- Composite number: has more than two positive factors.
- Number 1: neither prime nor composite.
- Number 2: the only even prime number.
- Divisibility rule: a shortcut for testing exact division.
- Prime factorisation: expressing a number as a product of prime numbers.
- Common factor: a factor shared by two or more numbers.
- Common multiple: a multiple shared by two or more numbers.