Finding the Highest Common Factor of Whole Numbers for JSS 1

Highest Common Factor (HCF)

Performance Objectives

By the end of the lesson, students should be able to:

1. Identify factors and common factors of whole numbers;
2. Find the HCF of whole numbers;
3. Identify the differences between LCM and HCF;
4. Apply the concept of HCF to real life;
5. Solve problems on quantitative aptitude tasks involving LCM and HCF of whole numbers.

Factors and Common Factors of Whole Numbers

A factor is a whole number that divides another whole number exactly without leaving a remainder.

Every non-zero whole number has a limited number of factors. The number 1 is a factor of every whole number, and every number is a factor of itself.

A common factor is a number that is a factor of two or more given numbers.

To find the factors of a number, multiplication or division facts can be used to identify all the whole numbers that can divide the target number without a remainder.

The following examples demonstrate how to find factors and common factors:

Worked Example 1

Find all the factors of 12.

To find the factors of 12, identify pairs of whole numbers that multiply together to give 12:

$$1 \times 12 = 12$$
$$2 \times 6 = 12$$
$$3 \times 4 = 12$$

The factors of 12 are: 1, 2, 3, 4, 6, and 12.

Worked Example 2

Find all the factors of 18.

Identify the pairs of whole numbers that multiply together to give 18:

$$1 \times 18 = 18$$
$$2 \times 9 = 18$$
$$3 \times 6 = 18$$

The factors of 18 are: 1, 2, 3, 6, 9, and 18.

Worked Example 3

Find the common factors of 12 and 18.

First, list the factors of each number:

Factors of 12 are: 1, 2, 3, 4, 6, 12

Factors of 18 are: 1, 2, 3, 6, 9, 18

Next, identify the numbers that appear in both lists:

The common factors of 12 and 18 are: 1, 2, 3, and 6.

Expressing Whole Numbers as Products of Prime Factors

A prime factor is a factor of a number that is also a prime number.

A prime number is a number greater than 1 that has only two factors: 1 and itself. Examples of prime numbers include 2, 3, 5, 7, 11, and 13. Any composite whole number can be written as a product of its prime factors. This process is called prime factorisation. Two common methods used to express a number as a product of prime factors are the factor tree method and the division method.

The Factor Tree Method

In this method, the original number is split into two branches representing any two factors. This process of branching continues for any composite factor until only prime numbers remain at the ends of all branches.

The diagram below illustrates the factor tree for the number 24.

From the factor tree, the prime factorisation of 24 is written as:

$$24 = 2 \times 2 \times 2 \times 3$$

Using index notation, this is expressed as:

$$24 = 2^3 \times 3$$

The Division Method

In this method, the given number is divided by the smallest prime number that can divide it exactly. The resulting quotient is then divided by the smallest prime number that can divide it, and the process continues until the quotient becomes 1.

The division method for the number 60 is shown below:

$$
\begin{array}{r|l}
2 & 60 \\
\hline
2 & 30 \\
\hline
3 & 15 \\
\hline
5 & 5 \\
\hline
& 1
\end{array}
$$

The prime factors of 60 are written as:

$$60 = 2 \times 2 \times 3 \times 5$$

Using index notation, this is expressed as:

$$60 = 2^2 \times 3 \times 5$$

Finding the Highest Common Factor (HCF)

The Highest Common Factor (HCF) is the largest common factor shared by two or more whole numbers.

The HCF is also sometimes referred to as the Greatest Common Divisor (GCD). There are two main methods used to find the HCF of whole numbers: the listing method and the prime factorisation method.

Method 1: The Listing Method

This method is useful for smaller numbers. It involves listing all the factors of each number, identifying the common factors, and selecting the largest one.

Worked Example 1

Find the HCF of 18 and 24 using the listing method.

First, list all the factors of each number:

Factors of 18 are: 1, 2, 3, 6, 9, 18

Factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24

Next, identify the common factors:

The common factors are: 1, 2, 3, and 6.

Identify the largest number in the list of common factors:

The largest common factor is 6.

Therefore, the HCF of 18 and 24 is 6.

Method 2: The Prime Factorisation Method

This method is highly efficient for larger numbers. It involves expressing each number as a product of its prime factors in index form, and then multiplying the lowest powers of the common prime factors.

Worked Example 2

Find the HCF of 24 and 36 using the prime factorisation method.

First, express each number as a product of prime factors:

$$24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3^1$$
$$36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2$$

Next, identify the common prime bases, which are 2 and 3.

Select the lowest power for each common prime base:

For base 2, the lowest power is $2^2$.

For base 3, the lowest power is $3^1$.

Multiply these lowest powers together to find the HCF:

$$\text{HCF} = 2^2 \times 3^1$$
$$\text{HCF} = 4 \times 3 = 12$$

Therefore, the HCF of 24 and 36 is 12.

Worked Example 3

Find the HCF of 24, 36, and 48.

First, express each of the three numbers as a product of prime factors in index form:

$$24 = 2^3 \times 3^1$$
$$36 = 2^2 \times 3^2$$
$$48 = 2^4 \times 3^1$$

Identify the common prime bases across all three numbers, which are 2 and 3.

Select the lowest power for each common prime base:

For base 2, the lowest power is $2^2$.

For base 3, the lowest power is $3^1$.

Multiply these values to obtain the HCF:

$$\text{HCF} = 2^2 \times 3^1$$
$$\text{HCF} = 4 \times 3 = 12$$

Therefore, the HCF of 24, 36, and 48 is 12.

Differences Between LCM and HCF

The Least Common Multiple (LCM) is the smallest non-zero common multiple of two or more numbers.

While HCF focuses on the largest common divisor (factor) that can divide the given numbers, LCM focuses on the smallest common multiple that the given numbers can divide into.

The table below outlines the key differences between LCM and HCF:

Worked Example

Find the HCF and LCM of 12 and 15 to compare their values.

First, express both numbers as products of prime factors:

$$12 = 2^2 \times 3^1$$
$$15 = 3^1 \times 5^1$$

To find the HCF, select the lowest power of the common prime factor (only 3 is common):

$$\text{HCF} = 3^1 = 3$$

To find the LCM, select the highest power of all prime factors present ($2$, $3$, and $5$):

$$\text{LCM} = 2^2 \times 3^1 \times 5^1$$
$$\text{LCM} = 4 \times 3 \times 5 = 60$$

This demonstrates that the HCF (3) is smaller than or equal to the given numbers, while the LCM (60) is larger than or equal to the given numbers.

Real-Life Applications of HCF

The concept of HCF is applied in various practical situations, such as simplifying fractions and solving grouping problems.

Application 1: Simplifying Fractions

To simplify a fraction to its lowest terms, divide both the numerator and the denominator by their HCF.

Worked Example 1

Simplify the fraction $\frac{24}{36}$ using the HCF.

First, find the HCF of the numerator (24) and the denominator (36). From previous calculations, the HCF of 24 and 36 is 12.

Divide both the numerator and the denominator by 12:

$$\frac{24 \div 12}{36 \div 12} = \frac{2}{3}$$

The simplified fraction is $\frac{2}{3}$.

Worked Example 2

Simplify the fraction $\frac{45}{60}$ using the HCF.

First, find the HCF of 45 and 60 using prime factorisation:

$$45 = 3^2 \times 5^1$$
$$60 = 2^2 \times 3^1 \times 5^1$$

The common prime factors are 3 and 5. The lowest power of 3 is $3^1$, and the lowest power of 5 is $5^1$.

$$\text{HCF} = 3^1 \times 5^1 = 15$$

Divide the numerator and the denominator by 15:

$$\frac{45 \div 15}{60 \div 15} = \frac{3}{4}$$

The simplified fraction is $\frac{3}{4}$.

Application 2: Word Problems

HCF is used to solve problems where objects must be divided into equal groups, rows, or packages without any leftovers.

Worked Example 3

A teacher has 20 notebooks and 30 pens. The teacher wants to distribute these items equally among a group of students so that no items are left over. What is the maximum number of students who can receive these items?

To find the maximum number of students, calculate the HCF of 20 and 30.

Express 20 and 30 as products of prime factors:

$$20 = 2^2 \times 5^1$$
$$30 = 2^1 \times 3^1 \times 5^1$$

Identify the common prime factors, which are 2 and 5. Select their lowest powers:

$$\text{HCF} = 2^1 \times 5^1 = 10$$

The maximum number of students who can receive the items is 10.

Each of the 10 students will receive:

$$20 \div 10 = 2 \text{ notebooks}$$
$$30 \div 10 = 3 \text{ pens}$$

Quantitative Aptitude Involving HCF and LCM

Quantitative aptitude tasks use visual patterns, shapes, or puzzles to test mathematical relationships and problem-solving skills.

In these puzzles, a specific rule connects the numbers. The task is to identify the rule and use it to find the missing number.

Aptitude Pattern 1: The Corner-to-Center Relationship

In this pattern, three numbers are placed at the corners of a triangle, and a single number is placed in the center. The center number is the HCF of the three corner numbers.

To solve the puzzle in the second triangle, find the HCF of the corner numbers 15, 25, and 35:

$$15 = 3^1 \times 5^1$$
$$25 = 5^2$$
$$35 = 5^1 \times 7^1$$

The only common prime factor is 5, and its lowest power is $5^1$.

$$\text{HCF} = 5$$

The missing number in the center of the second triangle is 5.

Aptitude Pattern 2: The Box Grid Relationship

In this pattern, a grid contains three boxes. The top box contains a value that represents the HCF of the two numbers in the bottom boxes.

Consider the completed pattern below:

Top Box: [ 8 ]

Bottom Left Box: [ 24 ] | Bottom Right Box: [ 40 ]

The relationship is verified because the HCF of 24 and 40 is 8.

Now, solve for the missing value $X$ in the following pattern:

Top Box: [ $X$ ]

Bottom Left Box: [ 18 ] | Bottom Right Box: [ 27 ]

To find $X$, calculate the HCF of 18 and 27:

$$18 = 2^1 \times 3^2$$
$$27 = 3^3$$

The common prime factor is 3, and its lowest power is $3^2$.

$$\text{HCF} = 3^2 = 9$$

Therefore, the value of $X$ is 9.

Questions for Evaluation

Solve the following problems to practice the concepts covered in this unit:

1. Find the HCF of the following sets of whole numbers:
   1. 16 and 24
   2. 15, 30, and 45
   3. 42 and 70
2. Express each of the following whole numbers as a product of its prime factors in index form:
   1. 72
   2. 100
   3. 120
3. Use the concept of HCF to simplify the following fractions to their lowest terms:
   1. $\frac{18}{30}$
   2. $\frac{48}{64}$
   3. $\frac{30}{75}$
4. A storekeeper wants to pack 36 tins of milk and 48 packets of biscuits into identical cartons. If every carton must contain the same number of each item with no items left over, what is the maximum number of cartons the storekeeper can pack?
5. Find the missing value $Y$ in the quantitative aptitude box pattern below:

   Top Box: [ $Y$ ]

   Bottom Left Box: [ 32 ] | Bottom Right Box: [ 48 ]