Highest Common Factor or HCF is an important arithmetic operation that plays an important part in adding, subtracting and comparing numerals. In this article, we will discuss HCF and the various ways to easily solve HCF problems.

Note that the Highest Common Factor or HCF is referred to as the Greatest Common Factor or GCF of two or more numbers. It is also referred to as Highest Common Multiple or Highest Common Divisor.

HIGHEST COMMON MULTIPLE (HCM)

The Highest Common Multiple of a number also refers to the HCF. The Highest Common Factor (HCF) can also be defined as the highest number that will be able to divide two or more numbers. For example, taking these three numbers 2, 7 and 14, these are common factors of 28, 42 and 70. 14 is the highest of the three numbers, therefore 14 is the HCF of 28, 42 and 70.

METHODS TO SOLVE HCF

Just like LCM, there are two methods of solving HCF:

1. Listing Method
2. Prime Factorization
3. Division method

LISTING METHOD

This is a method of solving HCF that involves listing the numbers that can divide the given numbers without leaving any remainder. In the listing method, the prime factors of the given numbers are listed, and then the Highest Common Factor is calculated by multiplying the common factors of the given numbers. Here are some exercises that use the listing method to find the HCF of given numbers.

1. Find the HCF of 20 and 30

Solution

20 = 1, 2, 4, 5, 10 and 20

30 = 1, 2, 3, 5, 6, 10, 15 and 30

The highest common number with the two numbers is 10. Therefore, the Highest Common Factor or HCF of 20 and 30 is 10.

2. Find the HCF of 24 and 30

Solution

24 = 1, 2, 3, 4, 6, 8, 12 and 24

30 = 1, 2, 3, 5, 6, 10, 15 and 30

The highest number common with the two numbers is 6. Therefore, the HCF of 24 and 30 is 6.

3. Find the HCF of 16 and 24

Solution

16 = 1, 2, 4, 8 and 16

24 = 1, 2, 3, 4, 6, 8, 12 and 24

Therefore, the Highest Common Factor or HCF is 8

4. Find the HCF OF 12 and 30

Solution

12 = 1, 2, 3, 4, 6 and 12

30 = 1, 2, 3, 5, 6, 10, 15 and 30

HCF = 6

5. Find the HCF of 30 and 40

Solution

30 = 1, 2, 3, 5, 6, 10, 15 and 30

40 = 1, 2, 4, 5, 8, 10, 20 and 40

HCF = 10

PRIME FACTORIZATION METHOD

To understand the prime factorization method we have to know how prime numbers and prime factors are used. A prime number is a whole number greater than 1 whose only component is 1 and itself. Prime factors of a number are all the numbers that can be multiplied to get the original number. 

In this method, the common prime factors are together to get the Highest Common Factor. Here are the steps to find the HCF using the prime factorization method:

• Write down the numbers as a product of the prime factor. This is known as prime factorization.
• Note the prime factors that are common to all the numbers.
• Multiply the common prime factors and note the product. 

The product is the Highest Common Factor or HCF. Here are some exercises that use the prime factorization method to find HCF.

1. Find the HCF of 20 and 30

Solution

20 = 2 x 2 x 5

30 = 2 x 3 x 5

The common prime factors of both numbers are 2 and 5. Their product is the HCF.

HCF = 2 x 5 = 10

2. Find the HCF of 24 and 30

Solution

24 = 2 x 2 x2 x 3

30 = 2 x 3 x 5

The common prime factors are 2 and 3.

HCF = 2 x 3 = 6

3. Find the HCF of 12, 24 and 48.

Solution

12 = 2 x 2 x 3

24 = 2 x 2 x 2 x 3

48 = 2 x 2 x 2 x 2 x 3 

The common prime factors are 2, 2 and 3.

Therefore, the HCF of 12, 16 and 48 = 2 x 2 x 3 = 12

DIVISION METHOD

This is another method of finding HCF by dividing the numbers by their common prime factors. The prime factors are then multiplied to get the Highest Common Factor. Below are a few applications of the division method in finding HCF.

1. Find the HCF of 20 and 30

Solution

The numbers are divided by common prime factors till they do not have a common prime factor anymore.

2 and 3 cannot be divided without remainder so we stop dividing there.

Therefore, the HCF of 20 and 30 is 2×5 = 10

2. Find the HCF of 24 and 30

Solution

4 and 5 cannot be divided without remainder so we stop division at this point.

Therefore, HCF of 24 and 30 is 2×3 = 6

3. Find the HCF of 16, 24 and 40.

Solution

2, 3 and 5 cannot be divided without remainder so we stop division at this point.

Therefore, the HCF of 16, 24, and 40 = 2x2x2 = 8

4. Find the HCF of 216, 288 and 360.

Solution

3, 4 and 5 cannot be divided without remainder, so we stop dividing there.

Therefore, the HCF of 216, 288 and 360 = 2 x 2 x 2 x 3 x 3 = 2³ x 9 = 72

INDEX NOTATION

As noticed in the last exercise, sometimes there is a series of the same number being multiplied by itself more than once. To represent this easily we use index notation where the number of times the number is multiplied by itself is shown as a superscript after the number. Here are examples of numbers and their index notations.

Considering 2², 2 is multiplied by itself like this 2 x 2 = 4

Considering 72, the prime factors = 2 x 2 x 2 x 3 x 3

To represent this in index notation, we count the number of times each number is repeated and write it after the number as a superscript like this:

72 = 2³ x 3² 

because 2 is multiplied by itself 3 times and 3 is multiplied by itself twice.

KEY POINTS

• HCF stands for Highest Common Factor.

• HCF (Highest Common Factor) is the same as HCM (Highest Common Multiple).

• Highest Common Factor (HCF) is the greatest of the prime factors common to two or more numbers.

• Index notation is very important in representing numbers as a product of prime factors.

• When using the product of the prime number method to calculate HCF, multiply the common multiples in the given numbers.

FAQ

1. How do I find the highest common factor?

Answer: You can find the Highest Common Factor by different methods but the most common are:

• Listing Method
• Prime Factorization Method
• Division Method

2. How do you calculate HCF fast?

Answer: To calculate HCF fast, you have to understand the different methods of calculations and find out the fastest way to solve it as different people prefer different methods.

3. Is HCF the same as GCF?

Answer: GCF stands for Greatest Common Factor. GCF is the same as the Highest Common Factor as both refer to the product of the common prime factors of given numbers.

Quiz

1. HCF stands for    

Answer: Highest Common Factor

2. Which of the following is the HCF of 12, 18 and 30?

1. 3
2. 6
3. 9
4. 12

Answer: B

3. What is 16 in index notation?

1. 2³ 
2. 2⁴
3. 2⁶
4. 2⁵

Answer: B