How to hcf of three numbers

What is the Highest Common Factor (HCF)?

The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of two or more integers is the largest positive integer that divides each of the integers without leaving a remainder. For example, the HCF of 12 and 18 is 6, because 6 is the largest number that divides both 12 (12 ÷ 6 = 2) and 18 (18 ÷ 6 = 3) evenly.

Methods to Find the HCF of Three Numbers

Finding the HCF of three numbers involves extending the methods used for two numbers. Here are the most common and effective ways:

Method 1: Listing Factors

This is a straightforward method, especially for smaller numbers. It involves listing all the factors (divisors) of each number and then identifying the largest factor that is common to all three lists.

Steps:

1. List all the factors of the first number.
2. List all the factors of the second number.
3. List all the factors of the third number.
4. Identify the common factors that appear in all three lists.
5. The largest of these common factors is the HCF.

Example: Find the HCF of 12, 18, and 30.

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 18: 1, 2, 3, 6, 9, 18
• Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30

The common factors are 1, 2, 3, and 6. The largest common factor is 6. Therefore, the HCF of 12, 18, and 30 is 6.

Method 2: Prime Factorization

This method is more systematic and efficient, especially for larger numbers. It involves breaking down each number into its prime factors and then multiplying the common prime factors raised to the lowest power they appear in any of the factorizations.

Steps:

1. Find the prime factorization of the first number.
2. Find the prime factorization of the second number.
3. Find the prime factorization of the third number.
4. Identify the prime factors that are common to all three factorizations.
5. For each common prime factor, take the lowest power it appears in any of the factorizations.
6. Multiply these common prime factors (with their lowest powers) together to get the HCF.

Example: Find the HCF of 24, 36, and 60.

• Prime factorization of 24: 2 × 2 × 2 × 3 = 2³ × 3¹
• Prime factorization of 36: 2 × 2 × 3 × 3 = 2² × 3²
• Prime factorization of 60: 2 × 2 × 3 × 5 = 2² × 3¹ × 5¹

The common prime factors are 2 and 3. The lowest power of 2 is 2² (from 36 and 60). The lowest power of 3 is 3¹ (from 24 and 60). The prime factor 5 is not common to all three. Therefore, the HCF is 2² × 3¹ = 4 × 3 = 12.

Method 3: Using the Euclidean Algorithm (Iterative Approach)

The Euclidean algorithm is primarily used for finding the HCF of two numbers. However, it can be extended to find the HCF of three numbers by applying it sequentially. This method is particularly efficient for very large numbers.

Steps:

1. Find the HCF of the first two numbers (let's say 'a' and 'b') using the Euclidean algorithm. Let this result be 'hcf1'.
2. Now, find the HCF of 'hcf1' and the third number ('c') using the Euclidean algorithm again.
3. The result of the second step is the HCF of all three numbers.

Euclidean Algorithm Recap (for two numbers x and y):

• If y = 0, then HCF(x, y) = x.
• Otherwise, HCF(x, y) = HCF(y, x mod y).

Example: Find the HCF of 48, 180, and 72.

1. Find HCF(48, 180):
• HCF(180, 48) = HCF(48, 180 mod 48) = HCF(48, 36)
• HCF(48, 36) = HCF(36, 48 mod 36) = HCF(36, 12)
• HCF(36, 12) = HCF(12, 36 mod 12) = HCF(12, 0)
• So, HCF(48, 180) = 12. Let hcf1 = 12.
2. Find HCF(hcf1, 72), which is HCF(12, 72):
• HCF(72, 12) = HCF(12, 72 mod 12) = HCF(12, 0)
• So, HCF(12, 72) = 12.

Therefore, the HCF of 48, 180, and 72 is 12.

Why is Finding the HCF Useful?

The HCF has practical applications in various areas of mathematics and daily life:

• Simplifying Fractions: Dividing both the numerator and denominator of a fraction by their HCF results in the simplest form of the fraction. For example, to simplify 24/36, the HCF is 12. Dividing both by 12 gives 2/3.
• Division into Equal Parts: When you need to divide quantities into the largest possible equal whole parts without any remainder, the HCF is used. For instance, if you have 12 apples and 18 oranges, the HCF (6) tells you that you can make 6 identical fruit baskets, each containing 2 apples and 3 oranges.
• Problem Solving: Many word problems in mathematics require finding the HCF to determine the largest possible size of identical items or groups.

Understanding how to calculate the HCF of three numbers is a fundamental skill in arithmetic that aids in simplifying calculations and solving a variety of practical problems.