LCM, HCF and Prime Numbers for Primary 4

LCM, HCF and Prime Numbers for Primary 4

Performance Objectives

By the end of the lesson, students should be able to:

1. find LCM of two to three 2-digit numbers;
2. find HCF of two to three 2-digit numbers;
3. describe prime number;
4. solve problems on quantitative reasoning.

Understanding Multiples

A multiple of a number is the result of multiplying that number by another whole number. Multiples are like the answers obtained when counting by a certain number.

For example, when counting by 2s, the numbers are 2, 4, 6, 8, 10, and so on. These are all multiples of 2.

Generating Multiples of Numbers

To generate multiples of a number, multiply the number by 1, then by 2, then by 3, and so on.

Examples of generating multiples include:

1. Multiples of 3:

To find the multiples of 3, multiply 3 by 1, 2, 3, 4, 5, and so on:

$3 \times 1 = 3$

$3 \times 2 = 6$

$3 \times 3 = 9$

$3 \times 4 = 12$

$3 \times 5 = 15$

The multiples of 3 are 3, 6, 9, 12, 15, …

2. Multiples of 5:

To find the multiples of 5, multiply 5 by 1, 2, 3, 4, 5, and so on:

$5 \times 1 = 5$

$5 \times 2 = 10$

$5 \times 3 = 15$

$5 \times 4 = 20$

$5 \times 5 = 25$

The multiples of 5 are 5, 10, 15, 20, 25, …

Least Common Multiple (LCM)

The Least Common Multiple (LCM) of two or more numbers is the smallest multiple that they have in common.

To find the LCM, list the multiples of each number until the first multiple that appears in all the lists is found.

Finding the LCM of Two 2-Digit Numbers

The procedure for finding the LCM of two 2-digit numbers involves the following steps:

1. List the multiples of the first number.
2. List the multiples of the second number.
3. Identify the common multiples (multiples that appear in both lists).
4. Choose the smallest common multiple. This is the LCM.

Worked Example 1: Finding the LCM of 12 and 18

Find the LCM of 12 and 18.

Step 1: List the multiples of 12:

Multiples of 12: 12, 24, 36, 48, 60, 72, 84, …

Step 2: List the multiples of 18:

Multiples of 18: 18, 36, 54, 72, 90, …

Step 3: Identify the common multiples:

Common multiples are 36, 72, …

Step 4: Choose the least common multiple:

The smallest common multiple is 36.

Therefore, the LCM of 12 and 18 is 36.

Worked Example 2: Finding the LCM of 15 and 25

Find the LCM of 15 and 25.

Step 1: List the multiples of 15:

Multiples of 15: 15, 30, 45, 60, 75, 90, …

Step 2: List the multiples of 25:

Multiples of 25: 25, 50, 75, 100, …

Step 3: Identify the common multiples:

The first common multiple is 75.

Step 4: Choose the least common multiple:

The smallest common multiple is 75.

Therefore, the LCM of 15 and 25 is 75.

Finding the LCM of Three 2-Digit Numbers

The procedure for finding the LCM of three 2-digit numbers involves the following steps:

1. List the multiples of the first number.
2. List the multiples of the second number.
3. List the multiples of the third number.
4. Identify the common multiples (multiples that appear in all three lists).
5. Choose the smallest common multiple. This is the LCM.

Worked Example 3: Finding the LCM of 10, 15, and 20

Find the LCM of 10, 15, and 20.

Step 1: List the multiples of 10:

Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, …

Step 2: List the multiples of 15:

Multiples of 15: 15, 30, 45, 60, 75, 90, …

Step 3: List the multiples of 20:

Multiples of 20: 20, 40, 60, 80, 100, …

Step 4: Identify the common multiples:

The first multiple that appears in all three lists is 60.

Step 5: Choose the least common multiple:

The smallest common multiple is 60.

Therefore, the LCM of 10, 15, and 20 is 60.

Understanding Factors

A factor of a number is a whole number that divides into it exactly, without leaving a remainder.

For example, the factors of 6 are 1, 2, 3, and 6 because each of these numbers divides 6 exactly.

Generating Factors of Numbers

To generate factors of a number, find all the pairs of numbers that multiply together to give that number.

Examples of generating factors include:

1. Factors of 12:

To find the factors of 12, think of pairs of numbers that multiply to 12:

$1 \times 12 = 12$

$2 \times 6 = 12$

$3 \times 4 = 12$

The factors of 12 are 1, 2, 3, 4, 6, and 12.

2. Factors of 20:

To find the factors of 20, think of pairs of numbers that multiply to 20:

$1 \times 20 = 20$

$2 \times 10 = 20$

$4 \times 5 = 20$

The factors of 20 are 1, 2, 4, 5, 10, and 20.

Highest Common Factor (HCF)

The Highest Common Factor (HCF) of two or more numbers is the largest factor that they have in common.

To find the HCF, list all the factors of each number and then find the largest factor that appears in all the lists.

Finding the HCF of Two 2-Digit Numbers

The procedure for finding the HCF of two 2-digit numbers involves the following steps:

1. List all the factors of the first number.
2. List all the factors of the second number.
3. Identify the common factors (factors that appear in both lists).
4. Choose the largest common factor. This is the HCF.

Worked Example 1: Finding the HCF of 24 and 36

Find the HCF of 24 and 36.

Step 1: List the factors of 24:

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Step 2: List the factors of 36:

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Step 3: Identify the common factors:

Common factors are 1, 2, 3, 4, 6, 12

Step 4: Choose the highest common factor:

The largest common factor is 12.

Therefore, the HCF of 24 and 36 is 12.

Worked Example 2: Finding the HCF of 40 and 60

Find the HCF of 40 and 60.

Step 1: List the factors of 40:

Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40

Step 2: List the factors of 60:

Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Step 3: Identify the common factors:

Common factors are 1, 2, 4, 5, 10, 20

Step 4: Choose the highest common factor:

The largest common factor is 20.

Therefore, the HCF of 40 and 60 is 20.

Finding the HCF of Three 2-Digit Numbers

The procedure for finding the HCF of three 2-digit numbers involves the following steps:

1. List all the factors of the first number.
2. List all the factors of the second number.
3. List all the factors of the third number.
4. Identify the common factors (factors that appear in all three lists).
5. Choose the largest common factor. This is the HCF.

Worked Example 3: Finding the HCF of 18, 27, and 45

Find the HCF of 18, 27, and 45.

Step 1: List the factors of 18:

Factors of 18: 1, 2, 3, 6, 9, 18

Step 2: List the factors of 27:

Factors of 27: 1, 3, 9, 27

Step 3: List the factors of 45:

Factors of 45: 1, 3, 5, 9, 15, 45

Step 4: Identify the common factors:

Common factors are 1, 3, 9

Step 5: Choose the highest common factor:

The largest common factor is 9.

Therefore, the HCF of 18, 27, and 45 is 9.

Prime Numbers

A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself.

This means a prime number can only be divided exactly by 1 and by the number itself.

Numbers that have more than two factors are called composite numbers.

The number 1 is neither prime nor composite because it only has one factor (itself).

Listing Prime Numbers

To list prime numbers, check each number to see how many factors it has.

Consider numbers from 2 to 20:

1. 2: Factors are 1, 2. (Exactly two factors). So, 2 is a prime number.
2. 3: Factors are 1, 3. (Exactly two factors). So, 3 is a prime number.
3. 4: Factors are 1, 2, 4. (More than two factors). So, 4 is a composite number.
4. 5: Factors are 1, 5. (Exactly two factors). So, 5 is a prime number.
5. 6: Factors are 1, 2, 3, 6. (More than two factors). So, 6 is a composite number.
6. 7: Factors are 1, 7. (Exactly two factors). So, 7 is a prime number.
7. 8: Factors are 1, 2, 4, 8. (More than two factors). So, 8 is a composite number.
8. 9: Factors are 1, 3, 9. (More than two factors). So, 9 is a composite number.
9. 10: Factors are 1, 2, 5, 10. (More than two factors). So, 10 is a composite number.
10. 11: Factors are 1, 11. (Exactly two factors). So, 11 is a prime number.
11. 12: Factors are 1, 2, 3, 4, 6, 12. (More than two factors). So, 12 is a composite number.
12. 13: Factors are 1, 13. (Exactly two factors). So, 13 is a prime number.
13. 14: Factors are 1, 2, 7, 14. (More than two factors). So, 14 is a composite number.
14. 15: Factors are 1, 3, 5, 15. (More than two factors). So, 15 is a composite number.
15. 16: Factors are 1, 2, 4, 8, 16. (More than two factors). So, 16 is a composite number.
16. 17: Factors are 1, 17. (Exactly two factors). So, 17 is a prime number.
17. 18: Factors are 1, 2, 3, 6, 9, 18. (More than two factors). So, 18 is a composite number.
18. 19: Factors are 1, 19. (Exactly two factors). So, 19 is a prime number.
19. 20: Factors are 1, 2, 4, 5, 10, 20. (More than two factors). So, 20 is a composite number.

The prime numbers between 1 and 20 are:

1. 2
2. 3
3. 5
4. 7
5. 11
6. 13
7. 17
8. 19

Quantitative Reasoning

Quantitative reasoning involves using numbers and mathematical operations to solve problems or understand patterns. It requires careful thinking about the relationships between numbers.

Solving Problems Using LCM, HCF, and Prime Numbers

Quantitative reasoning problems often require the application of knowledge of multiples, factors, LCM, HCF, and prime numbers.

Worked Example 1: Applying LCM

Two buses leave a park at the same time. Bus A returns to the park every 15 minutes, and Bus B returns every 20 minutes. After how many minutes will both buses be at the park again at the same time?

Solution:

This problem asks for a common time when both buses will be at the park. This means finding the Least Common Multiple (LCM) of 15 and 20.

Multiples of 15: 15, 30, 45, 60, 75, …

Multiples of 20: 20, 40, 60, 80, …

The LCM of 15 and 20 is 60.

Both buses will be at the park again at the same time after 60 minutes.

Worked Example 2: Applying HCF

A farmer has 30 oranges and 45 mangoes. He wants to pack them into baskets so that each basket has the same number of oranges and the same number of mangoes, without mixing fruits. What is the greatest number of baskets he can use?

Solution:

This problem asks for the greatest number that can divide both 30 and 45 exactly. This means finding the Highest Common Factor (HCF) of 30 and 45.

Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30

Factors of 45: 1, 3, 5, 9, 15, 45

The common factors are 1, 3, 5, 15.

The HCF of 30 and 45 is 15.

The farmer can use a maximum of 15 baskets.

Worked Example 3: Identifying Prime Numbers in a Sequence

Look at the sequence of numbers: 21, 23, 25, 27, 29. Which of these numbers are prime numbers?

Solution:

The factors for each number are checked:

1. 21: Factors are 1, 3, 7, 21. (More than two factors, so not prime).
2. 23: Factors are 1, 23. (Exactly two factors, so 23 is a prime number).
3. 25: Factors are 1, 5, 25. (More than two factors, so not prime).
4. 27: Factors are 1, 3, 9, 27. (More than two factors, so not prime).
5. 29: Factors are 1, 29. (Exactly two factors, so 29 is a prime number).

The prime numbers in the sequence are 23 and 29.