# Theory of Numbers Prime Factors, LCM and HCF and squares of numbers

**FIRST TERM **

**LEARNING NOTES**

**CLASS: JSS 2 (BASIC 8)**

**SCHEME OF WORK WITH LESSON NOTES **

**Subject: **

### MATHEMATICS

**Term:**

**FIRST TERM **

**Week:**

**WEEK 3**

**Class:**

**JSS 2 (BASIC 8)**

**Previous lesson: **

The pupils have previous knowledge of

that was taught as a topic during the last lesson.

**Topic :**

**Theory of Numbers Prime Factors, LCM and HCF **

**Behavioural objectives:**

At the end of the lesson, the pupils should be able to

- define LCM and HCF
- calculate the LCM and HCF of numbers
- Define and write out prime factors of numbers
- Calculate square and square roots of numbers

**Instructional Materials:**

- Wall charts
- Pictures
- Related Online Video
- Flash Cards

**Methods of Teaching:**

- Class Discussion
- Group Discussion
- Asking Questions
- Explanation
- Role Modelling
- Role Delegation

**Reference Materials:**

- Scheme of Work
- Online Information
- Textbooks
- Workbooks
- 9 Year Basic Education Curriculum
- Workbooks

**Content**

**Prime Factors of Numbers **

Prime numbers are figure with two prime numbers which are one and the number itself. Prime numbers have two factors, one and the number itself. Prime factors of a number are the factors of that number which are not divisible by any other number except that number and 1. For example, the prime factors of 30 are 2,3 and 5 because out of all the factors of 30 (1,2,3,5,6,10,15 and 30), it is only 2,3 and 5 that are not divisible by any other number except themselves and 1.

**Examples: **

Write the factors of the following numbers and state the prime factors, hence express each number as a product of its prime factors.

(a) 22 (b) 50 (c) 63 (d) 42

**Solutions: **

(a) 22

Factors of 22 are 1, 2, 11, 22

The prime factors of 22 are 2 and 11

Product of its prime factors of 22 = 2 × 11

(b) Factors of 50 are 1, 2, 5, 10, 25 and 50

The prime factors of 50 are 2 and 5

Product of its prime factors of 50 = 2 × 5 × 5 = 2 × 5^{2}

(c) Factors of 63 are 1, 3, 7, 9, 21 and 63

The prime factors of 63 are 3 and 7

Product of its prime factors of 63 = 3 × 3 × 7 = 3^{2} × 7

(d) Factors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42

The prime factors of 42 are 2, 3 and 7

Product of its prime factors of 42 = 2 × 3 × 7

** **

**Prime Factorization**

**Examples**

Express the following as a product of their prime factors in index form.

(a) 27 (b) 104 (c) 116

**Solutions: **

Divide each of the numbers by the prime factors in turns until it will not divide any further.

(a) 27

3 | 27 | |

3 | 9 | |

3 | 3 |

∴ 27 = 3 × 3 × 3 = 3^{3}

(b) 104

2 | 104 | |

2 | 52 | |

2 | 26 | |

13 | 13 | |

1 |

∴ 104 = 2 × 2 × 2 × 13 = 2^{3} × 13

(c) 116

2 | 116 | |

2 | 58 | |

29 | 29 | |

1 |

∴ 116 = 2 × 2 × 29 = 2^{2} × 29

**CLASS ACTIVITY**

- For each of the following numbers: 39, 53, 72 and 56

(a) Write their factors

(b) state which factors are prime numbers

(c) Express the numbers as product of its prime factors

- Show that 61 is a prime number
- Express each number as a product of its prime factors in index form: 117, 200, 98, 52 and 174.

**THE LEAST COMMON MULTIPLES (LCM) OF NUMBERS**

The LCM of two or more quantities is the smallest common multiple they have.

For example:

The multiple of 2: 2, 4, **6**, 8, 10, **12…**

The multiple of 3: 3, **6**, 9, **12**, 15…

Notice that 6 and 12 are the common multiples of 2 & 3

But the **Least Common multiple** (LCM) is 6.

Example: Find the LCM of 22, 30 and 40

Method: Express each number as a product of its prime factors.

22 = 2 × 11

30 = 2 × 3 × 5

40 = 2 × 2 × 2 × 5 = 2^{3} × 5

The prime factors of 22, 30 and 40 are 2, 3, 5, and 11.

The highest power of each prime factor must be in the LCM.

These are 2^{3}, 3, 5 and 11

Thus, LCM = 2^{3} × 3 × 5 × 11

= 8 × 3 × 5 × 11

= 1320

**Alternative method**

2 | 22 | 30 | 40 | |

2 | 11 | 15 | 20 | |

2 | 11 | 15 | 10 | |

3 | 11 | 15 | 5 | |

5 | 11 | 5 | 5 | |

11 | 11 | 1 | 1 | |

1 | 1 | 1 |

Thus, LCM = 2^{3} × 3 × 5 × 11 = 1320

** **

**CLASS ACTIVITY:- **Find the LCM of the following:

(a) 18, 30 and 48

(b) 8, 10 and 12

**THE HIGHEST COMMON FACTOR (HCF) OF NUMBERS**

14 is the highest common factor (HCF) of 28 and 42. It is the greatest number which will divide exactly into both 28 and 42.

**Example 1**

Find the HCF of 504 and 588.

**Method**: express each number as a product of its prime factors.

2 | 504 | |

2 | 252 | |

2 | 126 | |

3 | 63 | |

3 | 21 | |

7 | 7 | |

1 |

504 = 2^{3} ×3^{2} × 7

2 | 588 | |

2 | 294 | |

3 | 147 | |

7 | 49 | |

7 | 7 | |

1 |

588 = 2^{2} × 3 × 7^{2}

Find the common prime factors

504 = (2^{2} × 3 × 7) × 2 × 3

588 = (2^{2} × 3 × 7) × 7

The HCF is the product of the common prime factors.

HCF = 2^{2} × 3 x 7

= 4 × 3 × 7

= 84

** **

**CLASS ACTIVITY**

Find the HCF of the number 36, 54 and 60

** **

**SQUARE ROOTS OF PERFECT SQUARES BY FACTOR METHOD**

**SQUARE: **

7^{2}** = **7 × 7

In words ‘the square of 7 is 49’. We can turn this statement round and say, ‘the square root of 49 is 7’.

In symbols, 49−−√=7, to find the square root of a number, first find its factors.

**Examples**

(i) Find 11025−−−−−√.

Method: Try the prime number, first find its factors.

3 | 11025 | |

3 | 3675 | |

5 | 1225 | |

5 | 245 | |

7 | 49 | |

7 | 7 | |

1 |

11025 = 3^{2} × 5^{2} × 7^{2 }

= (3 × 5 × 7) × (3 × 5 × 7)

= 105 × 105

Thus 11025−−−−−√=105

It is not always necessary to write a number in its prime factors.

**Perfect Squares **

A perfect square is a whole number whose square root is also is also a whole number. i.e. 9, 25, 225, 9216 are perfect squares because their square roots are whole numbers.

It is possible to express a perfect square in factors with even indices. For example:

9216 = 96^{2 }= 3^{2} ×32^{2}

= 3^{2} × 4^{2} × 8^{2 }

= 3^{2} × 2^{10 }

(ii) Find the smallest number by which 540 must be multiplied so that the product is a perfect square.

2 | 540 | |

2 | 270 | |

3 | 135 | |

3 | 45 | |

3 | 15 | |

5 | 5 | |

1 |

540 = 2^{2} × 3^{3} × 5

The index of 2 is even.

The indices of 3 and 5 are odd.

one more 3 and one more 5 will make all the indices even. The product will then be a perfect square.

The number required = 3 × 5 = 15.

** **

**CLASS ACTIVITY**

- Find by factors the square roots of the following: (i) 484 (ii) 2500
- Find the smallest numbers by which the following must be multiplied so that their products are perfect squares. (a) 2 × 2 × 2 × 3 × 3 × 5

(b) 2^{3} × 3^{4} × 5^{6} × 7^{2} × 13^{2}

**QUANTITATIVE REASONING**

Sample A 4, 8, 12, 16, 20,…

2, 4, 6, 8, 10,…

6, 12, 18, 24, 30, …

Supply the missing numbers in 16, 20, ……, 28, …….

The set are multiples of 4 because 4 can divide all the given numbers without remainder, so successive addition of 4 gives 16 + 4 = 20, 20+ 4 = 24; 24 + 4 = 28; 28+ 4 = 32

Fill in the gaps.

(i) 15, 30, ……., 60, ……..

(ii) 4, 6, ……., ……., 12, 14

**Sample B**

**PRACTICE QUESTIONS**

- Find the L.C.M of the following:

(a) 12, 15 and 18

(b) 20, 28 and 30

(c) 216 and 16

- Find the HCF of the following:

(a) 72, 108 and 54

(b) 36, 54 and 60

(c) 54 and 105

- Find the factors of the square roots of the following:

(a) 1 936

(b) 2 025

(c) 2 916

(d) 3 136

(e) 1 225

- What must be multiplied by 240 to make it a perfect square?
- Express 11 664 as a product of its factors in index form

** **

**Presentation**

The topic is presented step by step

Step 1:

The class teacher revises the previous topics

Step 2.

He introduces the new topic

Step 3:

The class teacher allows the pupils to give their own examples and he corrects them when the needs arise

### Evaluation

**ASSIGNMENT**

- Using the product of prime factors, find the square root of (i) 400 (ii) 750
- What is the smallest number by which 90 must be multiplied to obtain a perfect square?
- Find the square root of the following by factorization method

(a) 5 184

(b) 46 565

(c) 7 744

(d) 2 916

(e) 41 209

- A square plank has an area of 116.64cm
^{2}. What is the length of its sides? - A pile of nuts can be shared equally between 8 boys, between 9 boys and between 10 boys. What is the least number of nuts in the pile?
- How many five-digit numbers are divisible by 11, If the first digit is 1, the third digit is 3 and the fifth digit is 5?
- Express 38 as a product of two prime numbers.
- Factorize and so find the square root of 64, 81, 144, 256, 729, 1600

**Conclusion**

The class teacher wraps up or concludes the lesson by giving out a short note to summarize the topic that he or she has just taught.

The class teacher also goes round to make sure that the notes are well copied or well written by the pupils.

He or she makes the necessary corrections when and where the needs arise.

** **