Specific Objectives
By the end of the topic, the learner should be able to test the divisibility of numbers by 2, 3, 4, 5, 6, 8, 9, 10, and 11.
Content
Divisibility tests for numbers by 2, 3, 4, 5, 6, 8, 9, 10, and 11.
Introduction
Divisibility tests simplify computations involving numbers. The following table summarizes the rules for divisibility by various numbers.
| Divisibility Tests | Example |
|---|---|
| A number is divisible by 2 if the last digit is 0, 2, 4, 6, or 8. | 168 is divisible by 2 since the last digit is 8. |
| A number is divisible by 3 if the sum of its digits is divisible by 3. | 168 is divisible by 3 since the sum of the digits is 15 (1 + 6 + 8 = 15), and 15 is divisible by 3. |
| A number is divisible by 4 if the number formed by the last two digits is divisible by 4. | 316 is divisible by 4 since 16 is divisible by 4. |
| A number is divisible by 5 if the last digit is either 0 or 5. | 195 is divisible by 5 since the last digit is 5. |
| A number is divisible by 6 if it is divisible by both 2 and 3. | 168 is divisible by 6 since it is divisible by 2 and 3. |
| A number is divisible by 8 if the number formed by the last three digits is divisible by 8. | 7,120 is divisible by 8 since 120 is divisible by 8. |
| A number is divisible by 9 if the sum of its digits is divisible by 9. | 549 is divisible by 9 since the sum of the digits is 18 (5 + 4 + 9 = 18), and 18 is divisible by 9. |
| A number is divisible by 10 if the last digit is 0. | 1,470 is divisible by 10 since the last digit is 0. |
| A number is divisible by 11 if the difference between the sum of its digits in the odd positions (1st, 3rd, 5th, 7th, etc.) and the sum of its digits in the even positions (2nd, 4th, 6th, 8th, etc.) is either 0 or a multiple of 11. | 8,260,439: sum of digits in odd positions = 8 + 6 + 4 + 9 = 27; sum of digits in even positions = 2 + 0 + 3 = 5; difference = 27 – 5 = 22, which is a multiple of 11. |
End of Topic
Did you understand everything? If not, ask a teacher, friends, or anybody, and make sure you understand before going to sleep! |
Past KCSE Questions on the Topic
CHAPTER FOUR
Specific Objectives
By the end of the topic, the learner should be able to:
- Find the GCD/HCF of a set of numbers.
- Apply GCD to real-life situations.
Content
- GCD of a set of numbers
- Application of GCD/HCF to real-life situations
Introduction
The Greatest Common Divisor (GCD) is the largest number that is a factor of two or more numbers.
When finding the Greatest Common Factor, you look for the common factors shared by the given numbers. To find the GCD of two or more numbers, first list the factors of each number, identify the common factors, and then state the greatest among them.
The GCD can also be found by expressing each number as a product of its prime factors. The common prime factors are then identified, and their product gives the GCD.
Example
Find the Greatest Common Factor (GCD) of 36 and 54.
Solution
The prime factorization of 36 is 2 × 2 × 3 × 3.
The prime factorization of 54 is 2 × 3 × 3 × 3.
They have common factors 2, 3, and 3, and their product is 18.
Therefore, the greatest common factor of 36 and 54 is 18.
Example
Find the GCD of 72, 96, and 300.
Solution
| 72 | 96 | 300 | |
| 2 | 36 | 48 | 150 |
| 2 | 18 | 24 | 75 |
| 3 | 6 | 8 | 25 |
End of Topic
Did you understand everything? If not, ask a teacher, friends, or anybody, and make sure you understand before going to sleep! |
Past KCSE Questions on the Topic
- Find the greatest common divisor of the terms 144x3y2 and 81xy4.
b) Hence, factorize completely the expression 144x3y2 – 81xy4. (2 marks)
- The GCD of two numbers is 7 and their LCM is 140. If one of the numbers is 20, find the other number.
- The LCM of three numbers is 7920 and their GCD is 12. Two of the numbers are 48 and 264. Using factor notation, find the third number if one of its factors is 9.
