Highest Common Factor (hcf)

Notes

The highest common factor (HCF) is defined by the largest integer shared by two or more numbers, e.g., the HCF of 20 and 40 is 20.

The most efficient method for calculating the HCF would involve drawing tree diagrams and a Venn diagram. Follow the steps outlined below:

Step 1) Draw the tree diagram (of the given numbers) as a product of its prime factors, e.g., 56 as a product of its prime factors is written as 7 * 2 * 2 * 2

NOTE: A prime number is a number that produces an integer output only when it is divided by itself and 1. Therefore, when a given number is written as a product of its prime factors, by definition, they cannot be simplified to be written as a product of more factors.

Step 2) Place each factor in the correct part of the Venn diagram with common factors (between the given numbers) being placed in the intersection of the circles drawn

Question 1) Find the Highest Common Factor of 12 and 112

Question 2) Compute the Highest Common Factor of 39 and 85

Question 3) Work out the Highest Common Factor of 40 and 96

Question 4) Find the Highest Common Factor of 252 and 155

Answer 1)

12 = 4 * 3 = 2 * 2 * 3 = 3 * 2²

112 = 2 * 56 = 2 * 2 * 28 = 2 * 2 * 4 * 7 = 2 * 2 * 2 * 2 * 7 = 7 * 2⁴

So, HCF = &2 * 2 = 4

Answer 2)

39 = 3 * 13

85 = 5 * 17

So, HCF = 1 (since there are NO common prime factors between 39 and 85)

Answer 3)

40 = 5 * 8 = 5 * 4 * 2 = 5 * 2 * 2 * 2 = 2³ * 5

96 = 2 * 48 = 2 * 2 * 24 = 2 * 2 * 2 * 12 = 2 * 2 * 2 * 2 * 6 = 2 * 2 * 2 * 2 * 2 * 3 = 2⁵ * 3

So, HCF = 2 * 2 * 2 = 8

Answer 4)

252 = 2 * 126 = 2 * 2 * 63 = 2 * 2 * 21 * 3 = 2 * 2 * 7 * 3 * 3 = 3² * 2² * 7

155 = 5 * 31

So, HCF = 1

Answer 5)

HCF of x and 6x = x