Chapter 1.7 (Mathematics 5)

Highest common factors (HCF) and greatest common divisors (GCD)

Factors and divisors

Group Activity 14

In pairs:

  1. Write pairs of numbers whose product is 18.
  2. List in order the numbers that appear in the product.
  3. Discuss and share your results with other groups

Note

  1. The pair of numbers whose product is 18 are the factors of 18.
  2. The list of numbers involved in the multiplication to get 18 are the divisors of 18.
  3. The factors of a number are also the divisors of the number.

Example 14

Find:
  1. the pair of factors of 24
  2. the divisors of 24
  1. 24 is 1 × 24, 2 × 12, 3 × 8, 4 × 6
    ​The pair of factors of 24 are:
    ​(1, 24) (2, 12), (3, 8) and (4, 6)
  2. The divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.

Exercise Q

(1, )

(, 5)

  • 1
  • 2
  • 3
  • 4
  • 5
  • 6
  • 7
  • 8
  • 9
  • 10
  • 11
  • 14
  • 13
  • 14
  • 15
  • 16
  1. 8
  2. 12
  3. 32
  4. 48
  1. 10
  2. 28
  3. 54
  4. 75

Q5

Kunitu arranged 12 bottle tops as shown.

In pairs, arrange 12 bottle tops in other different rectangular patterns.

Common factors and common divisors

Group Activity 15

In pairs:

  1. Pick any two numbers between 10 and 30.
  2. List the factors or divisors of each number.
  3. Find the common factors and common divisors of the two numbers.
  4. Discuss and share the results with other groups.

Example 15

Find the common factors or common divisors of 24 and 32.
  • Factors of 24 are (1, 24) (2, 12), (3, 8) and (4, 6). So factors or divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
  • Factors or divisors of 32 are 1, 2, 4, 8, 16 and 32.
  • Common factors or divisors of 24 and 32 are 1, 2, 4 and 8.

Exercise R

R1

Find the common factors of:

  1. 42 and 56
  2. 48 and 60

R2

Find the common divisors of:

  1. 21 and 35
  2. 28 and 48

R3

Find the common factors or divisors of:

  1. 8, 12 and 18
  2. 12, 30 and 36

Digital Time

Common factors or common divisors

Highest common factors and greatest common divisors

Group Activity 16

In groups of three:

  1. Take two numbers between 20 and 30.
  2. List the factors or divisors of each number.
  3. Identify and list the common factors or common divisors of the two numbers.
  4. Identify and write the highest of the common factors of the numbers.
  5. Discuss the results and share with other groups.

Note

The Highest Common Factor (HCF) is also the Greatest Common Divisor (GCD).

Example 16

Find the GCD (HCF) of 18 and 24.
  • Divisors of 18 are 1, 2, 3, 6, 9 and 18.
  • Divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
  • Common divisors are 1, 2, 3 and 6.
  • The Greatest Common Divisor (GCD) is 6.

Exercise S

The HCF is .

The GCD is .

S3

Find the HCF of:

  1. 12, 16
  2. 18, 32
  1. 48, 54
  2. 18, 24, 36

S4

Find the greatest number that can divide 45 and 75 without a remainder.

S5

Find the largest possible number of pupils who can share 56 exercise books and 64 pencils equally.

S6

Gitu bought 12 bananas and apples. She shared the fruits equally among his children. Find the largest possible number of children who shared the fruits.

S7

Umoja bought 30 cows and 40 goats. He shared the animals among some farmers. Find the highest possible number of farmers who shared the animals equally.

S8

Omari bought 36 mangoes and 48 sweets for his birthday party. He shared the items equally among the people who attended the party. What was the largest possible number of people who attended the party?

Digital Time

Highest common fractor