HCF of Two Numbers

The highest common factor (HCF) of two or more numbers is the largest positive integer that divides each of the numbers without leaving a remainder. It is also known as the greatest common divisor (GCD).

Finding the HCF

There are several methods for finding the HCF of two numbers. One common method is the Euclidean algorithm, which involves repeatedly dividing the larger number by the smaller number and taking the remainder. The last non-zero remainder is the HCF.

For example, to find the HCF of 12 and 18, we can use the Euclidean algorithm as follows:

1. Divide 18 by 12: 18 ÷ 12 = 1 remainder 6
2. Divide 12 by 6: 12 ÷ 6 = 2 remainder 0

The last non-zero remainder is 6, so the HCF of 12 and 18 is 6.

Properties of the HCF

The HCF of two numbers has several important properties, including:

• The HCF of two numbers is always a positive integer.
• The HCF of two numbers is a divisor of both numbers.
• The HCF of two numbers is unique.
• The HCF of two numbers is equal to the product of their common prime factors.

Applications of the HCF

The HCF of two numbers has several applications, including:

• Simplifying fractions: The HCF of the numerator and denominator of a fraction can be used to simplify the fraction.
• Finding the least common multiple (LCM): The LCM of two numbers is the smallest positive integer that is divisible by both numbers. The LCM of two numbers can be found by multiplying their HCF by their product.
• Solving Diophantine equations: Diophantine equations are equations that have integer solutions. The HCF of the coefficients of a Diophantine equation can be used to find its solutions.

The HCF of two numbers is a fundamental concept in number theory with various applications. It can be found using the Euclidean algorithm and has several important properties.

Methods to find HCF of Two Numbers

The highest common factor (HCF) of two numbers is the largest positive integer that divides both numbers without leaving a remainder. There are several methods to find the HCF of two numbers. Here are two commonly used methods:

1. Prime Factorization Method

This method involves finding the prime factors of both numbers and then identifying the common prime factors. The product of these common prime factors is the HCF of the two numbers.

Steps:

1. Write down the prime factorization of each number.
2. Identify the common prime factors.
3. Multiply the common prime factors to find the HCF.

Example:

Find the HCF of 12 and 18.

Solution:

1. Prime factorization of 12: 2 x 2 x 3
2. Prime factorization of 18: 2 x 3 x 3
3. Common prime factors: 2 and 3
4. HCF of 12 and 18 = 2 x 3 = 6

2. Euclidean Algorithm

The Euclidean algorithm is a more efficient method for finding the HCF of two numbers. It involves repeatedly dividing the larger number by the smaller number and taking the remainder. The last non-zero remainder is the HCF of the two numbers.

Steps:

1. Divide the larger number by the smaller number and find the remainder.
2. Repeat step 1 with the previous divisor and the remainder until the remainder becomes 0.
3. The last non-zero remainder is the HCF of the two numbers.

Example:

Find the HCF of 12 and 18 using the Euclidean algorithm.

Solution:

1. 18 ÷ 12 = 1 remainder 6
2. 12 ÷ 6 = 2 remainder 0
3. The last non-zero remainder is 6.
4. Therefore, the HCF of 12 and 18 is 6.

The Euclidean algorithm is particularly useful when dealing with large numbers, as it does not require finding the prime factors of the numbers.

HCF of two numbers Special Case:

The highest common factor (HCF) of two numbers is the largest positive integer that divides both numbers without leaving a remainder. In other words, it is the greatest number that is a factor of both numbers.

Special Case: One Number is Zero

If one of the two numbers is zero, then the HCF is the other number. This is because any number divided by zero is undefined, so the only number that divides both zero and another number is the other number itself.

For example, the HCF of 12 and 0 is 12, and the HCF of 0 and -7 is -7.

Proof

Let’s consider two numbers, $a$ and $b$, where $a$ is not equal to zero. We can write $b$ as a multiple of $a$, plus a remainder:

$$b = aq + r$$

where $q$ is the quotient and $r$ is the remainder.

If $r$ is equal to zero, then $b$ is divisible by $a$, and $a$ is the HCF of $a$ and $b$.

If $r$ is not equal to zero, then we can repeat the process with $a$ and $r$. We can write $a$ as a multiple of $r$, plus a remainder:

$$a = rq’ + r’$$

where $q’$ is the quotient and $r’$ is the remainder.

If $r’$ is equal to zero, then $r$ is the HCF of $a$ and $b$.

If $r’$ is not equal to zero, then we can repeat the process again. We will eventually reach a point where the remainder is zero, and the last non-zero remainder will be the HCF of $a$ and $b$.

Example

Let’s find the HCF of 12 and 18 using the Euclidean algorithm.

$$18 = 12 \cdot 1 + 6$$

$$12 = 6 \cdot 2 + 0$$

Since the last non-zero remainder is 6, the HCF of 12 and 18 is 6.

HCF of Two Numbers Solved Examples

The highest common factor (HCF) of two or more numbers is the largest positive integer that divides each of the numbers without leaving a remainder.

Example 1: Finding the HCF of 12 and 18

To find the HCF of 12 and 18, we can use the following steps:

1. List the factors of each number.

   12: 1, 2, 3, 4, 6, 12 18: 1, 2, 3, 6, 9, 18

2. Identify the common factors of both numbers.

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

3. The HCF of 12 and 18 is the largest of the common factors, which is 6.

Example 2: Finding the HCF of 24, 36, and 48

To find the HCF of 24, 36, and 48, we can use the following steps:

1. List the factors of each number.

   24: 1, 2, 3, 4, 6, 8, 12, 24 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

2. Identify the common factors of all three numbers.

The common factors of 24, 36, and 48 are 1, 2, 3, 4, 6, and 12.

3. The HCF of 24, 36, and 48 is the largest of the common factors, which is 12.

Example 3: Finding the HCF of 100 and 120

To find the HCF of 100 and 120, we can use the following steps:

1. List the factors of each number.

   100: 1, 2, 4, 5, 10, 20, 25, 50, 100 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120

2. Identify the common factors of both numbers.

The common factors of 100 and 120 are 1, 2, 4, 5, 10, and 20.

3. The HCF of 100 and 120 is the largest of the common factors, which is 20.

Conclusion

The HCF of two or more numbers can be found by listing the factors of each number, identifying the common factors, and then selecting the largest of the common factors.

HCF of Two Numbers FAQs

What is the HCF of two numbers?

The HCF (Highest Common Factor) of two numbers is the largest positive integer that divides both numbers without leaving a remainder.

How to find the HCF of two numbers?

There are several methods to find the HCF of two numbers. Some common methods include:

• Prime Factorization Method: This method involves expressing both numbers as products of their prime factors and then identifying the common prime factors. The product of these common prime factors is the HCF.

• Euclidean Algorithm: This is an efficient algorithm for finding the HCF of two numbers. It involves repeatedly dividing the larger number by the smaller number and taking the remainder. The last non-zero remainder is the HCF.

What is the HCF of two co-prime numbers?

Two numbers are said to be co-prime (or relatively prime) if they have no common factors other than 1. The HCF of two co-prime numbers is 1.

What is the HCF of two consecutive numbers?

The HCF of two consecutive numbers is 1.

What is the HCF of a number and 0?

The HCF of a number and 0 is the number itself.

What is the HCF of a number and 1?

The HCF of a number and 1 is 1.

What is the HCF of a number and itself?

The HCF of a number and itself is the number itself.

What is the HCF of two negative numbers?

The HCF of two negative numbers is the positive value of the HCF of their absolute values.

What is the HCF of two fractions?

To find the HCF of two fractions, first convert them to improper fractions. Then, find the HCF of the numerators and denominators separately. The HCF of the numerators is the numerator of the HCF, and the HCF of the denominators is the denominator of the HCF.

What is the HCF of three or more numbers?

To find the HCF of three or more numbers, first find the HCF of two numbers. Then, find the HCF of the result and the third number. Repeat this process until all numbers are considered. The final result is the HCF of all the given numbers.