HCF and LCM

The highest common factor (HCF) of two or more numbers is the largest number that divides each of the numbers without leaving a remainder. The least common multiple (LCM) of two or more numbers is the smallest number that is divisible by each of the numbers.

To find the HCF of two numbers, you can use the Euclidean algorithm. This algorithm involves repeatedly dividing the larger number by the smaller number and taking the remainder. The last non-zero remainder is the HCF.

To find the LCM of two numbers, you can use the formula LCM = $\dfrac{(a * b)} {HCF}$, where a and b are the two numbers.

The HCF and LCM are important concepts in number theory and have many applications, such as finding the greatest common divisor of two polynomials, simplifying fractions, and solving systems of linear equations.

HCF and LCM Definition

HCF (Highest Common Factor)

The HCF (Highest Common Factor) of two or more numbers is the largest positive integer that is a factor of all the given numbers. In other words, it is the greatest number that divides all the given numbers without leaving a remainder.

Example:

Find the HCF of 12, 18, and 24.

Solution:

The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 18 are 1, 2, 3, 6, 9, and 18. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

The common factors of 12, 18, and 24 are 1, 2, 3, and 6.

The HCF of 12, 18, and 24 is 6.

LCM (Least Common Multiple)

The LCM (Least Common Multiple) of two or more numbers is the smallest positive integer that is divisible by all the given numbers. In other words, it is the least number that is a multiple of all the given numbers.

Example:

Find the LCM of 12, 18, and 24.

Solution:

The multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, 108, and so on. The multiples of 18 are 18, 36, 54, 72, 90, 108, 126, 144, 162, and so on. The multiples of 24 are 24, 48, 72, 96, 120, 144, 168, 192, 216, and so on.

The common multiples of 12, 18, and 24 are 72, 144, 216, 288, 360, and so on.

The LCM of 12, 18, and 24 is 72.

Relationship between HCF and LCM

The HCF and LCM of two or more numbers are related by the following formula:

HCF × LCM = Product of the numbers

Example:

Find the HCF and LCM of 12 and 18.

Solution:

The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 18 are 1, 2, 3, 6, 9, and 18.

The HCF of 12 and 18 is 6.

The multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, 108, and so on. The multiples of 18 are 18, 36, 54, 72, 90, 108, 126, 144, 162, and so on.

The LCM of 12 and 18 is 36.

The product of 12 and 18 is 216.

HCF × LCM = 6 × 36 = 216

Therefore, the HCF and LCM of 12 and 18 satisfy the formula HCF × LCM = Product of the numbers.

How to find HCF and LCM?

How to Find the Highest Common Factor (HCF) and Least Common Multiple (LCM)

The highest common factor (HCF) of two or more numbers is the largest number that divides each of the numbers without leaving a remainder. The least common multiple (LCM) of two or more numbers is the smallest number that is divisible by each of the numbers.

Finding the HCF

There are two common methods for finding the HCF of two or more numbers: the prime factorization method and the Euclidean algorithm.

Prime Factorization Method

1. Write each number as a product of its prime factors.
2. Identify the common prime factors.
3. Multiply the common prime factors together.

Example:

Find the HCF of 12 and 18.

1. 12 = 2 * 2 * 3
2. 18 = 2 * 3 * 3
3. The common prime factors are 2 and 3.
4. 2 * 3 = 6

Therefore, the HCF of 12 and 18 is 6.

Euclidean Algorithm

1. Divide the larger number by the smaller number.
2. Take the remainder and divide it into the previous divisor.
3. Repeat steps 1 and 2 until the remainder is 0.
4. The last non-zero remainder is the HCF.

Example:

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

1. 18 ÷ 12 = 1 remainder 6
2. 12 ÷ 6 = 2 remainder 0

Therefore, the HCF of 12 and 18 is 6.

Finding the LCM

The LCM of two or more numbers is the smallest number that is divisible by each of the numbers. There are two common methods for finding the LCM of two or more numbers: the prime factorization method and the least common multiple formula.

Prime Factorization Method

1. Write each number as a product of its prime factors.
2. Identify the common prime factors and the non-common prime factors.
3. Multiply the common prime factors together.
4. Multiply the non-common prime factors together.
5. The product of the common and non-common prime factors is the LCM.

Example:

Find the LCM of 12 and 18.

1. 12 = 2 * 2 * 3
2. 18 = 2 * 3 * 3
3. The common prime factors are 2 and 3.
4. The non-common prime factors are 2 and 3.
5. 2 * 2 * 3 * 3 = 36

Therefore, the LCM of 12 and 18 is 36.

Least Common Multiple Formula

The LCM of two numbers can also be found using the following formula:

$ LCM = \dfrac{(a * b)} {HCF} $

where a and b are the two numbers.

Example:

Find the LCM of 12 and 18 using the least common multiple formula.

LCM = $\dfrac{(12 * 18)} {6}$

LCM = 36

Therefore, the LCM of 12 and 18 is 36.

Frequently Asked Questions on HCF and LCM

What is the full form of HCF in Maths? Explain HCF with an example.

What is the full form of LCM in Maths? Explain LCM with an example.

LCM Full Form in Maths

The full form of LCM in maths is the Least Common Multiple. It is the smallest positive integer that is divisible by two or more given integers without leaving a remainder.

Example of LCM

Find the LCM of 2, 3, and 4.

1. List the multiples of each number:

2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, …

3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, …

4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …

2. Identify the common multiples of 2, 3, and 4:

12, 24, 36, …

3. The LCM of 2, 3, and 4 is the smallest common multiple, which is 12.

Properties of LCM

The LCM of two or more integers has the following properties:

1. The LCM of two or more integers is always a positive integer.
2. The LCM of two or more integers is divisible by each of the given integers.
3. The LCM of two or more integers is the smallest positive integer that is divisible by each of the given integers.
4. The LCM of two or more integers can be found by multiplying the prime factors of each integer and taking the highest power of each prime factor.

Applications of LCM

The LCM is used in a variety of mathematical applications, including:

1. Finding the least common denominator of fractions
2. Solving systems of linear equations
3. Finding the period of a repeating decimal
4. Finding the greatest common divisor of two or more integers

What is the GCD of 24 and 36?

The greatest common divisor (GCD) of two or more numbers is the largest positive integer that is a factor of all of the given numbers. In other words, it is the largest number that divides each of the given numbers without leaving a remainder.

To find the GCD of 24 and 36, we can list the factors of each number and identify the largest common factor.

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

Factors of 36 : 1, 2, 3, 4, 6, 9, 12, 18, 36

The largest common factor of 24 and 36 is 12.

Here is another example:

What is the GCD of 12, 18, and 24?

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

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

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

The largest common factor of 12, 18, and 24 is 6.

We can also use a mathematical formula to find the GCD of two or more numbers. The formula is:

$GCD(a, b) = \dfrac {a × b} {LCM(a, b)}$

where a and b are the given numbers and LCM(a, b) is the least common multiple of a and b.

For example, the GCD of 24 and 36 can be found using the formula:

$GCD(24, 36) = \dfrac{24 × 36} {LCM(24, 36)}$

First, we need to find the LCM of 24 and 36. The LCM is the smallest positive integer that is divisible by both 24 and 36.

To find the LCM, we can list the multiples of each number and identify the smallest common multiple.

Multiples of 24 : 24, 48, 72, 96, 120, …

Multiples of 36 : 36, 72, 108, 144, 180, …

The smallest common multiple of 24 and 36 is 72.

Now we can substitute the values of a, b, and LCM(a, b) into the formula:

$GCD(24, 36) = {24 × 36} {72}$

$GCD(24, 36) = 12$

Therefore, the GCD of 24 and 36 is 12.

What is the formula for HCF and LCM?

Highest Common Factor (HCF) or Greatest Common Divisor (GCD):

The HCF of two or more numbers is the largest positive integer that divides each of the given numbers without leaving a remainder.

Formula for HCF:

The HCF of two numbers, a and b, can be found using the Euclidean algorithm:

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.

1. 18 ÷ 12 = 1 remainder 6
2. 12 ÷ 6 = 2 remainder 0

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

Least Common Multiple (LCM):

The LCM of two or more numbers is the smallest positive integer that is divisible by each of the given numbers.

Formula for LCM:

The LCM of two numbers, a and b, can be found using the formula:

$LCM(a, b) = \dfrac{(a \times b)} {HCF(a, b)}$

Example:

Find the LCM of 12 and 18.

1. $HCF(12, 18) = 6$
2. $LCM(12, 18) = \dfrac{(12 \times 18)} { 6} = 36$

Therefore, the LCM of 12 and 18 is 36.

How can we find the LCM and HCF?

LCM (Least Common Multiple):

The LCM of two or more numbers is the smallest positive integer that is divisible by all the given numbers without leaving a remainder.

Example:

Find the LCM of 2, 3, and 4.

1. List the multiples of each number:

• Multiples of 2 : 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, …
• Multiples of 3 : 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, …
• Multiples of 4 : 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …

2. Identify the common multiples:

• The common multiples of 2, 3, and 4 are 12, 24, 36, 48, 60, …

3. The LCM of 2, 3, and 4 is the smallest common multiple, which is 12.

HCF (Highest Common Factor):

The HCF of two or more numbers is the largest positive integer that is a factor of all the given numbers.

Example:

Find the HCF of 12, 18, and 24.

1. List the factors of each number:

• Factors of 12 : 1, 2, 3, 4, 6, 12
• Factors of 18 : 1, 2, 3, 6, 9, 18
• Factors of 24 : 1, 2, 3, 4, 6, 8, 12, 24

2. Identify the common factors:

• The common factors of 12, 18, and 24 are 1, 2, 3, 6.

3. The HCF of 12, 18, and 24 is the largest common factor, which is 6.

Note:

• The LCM and HCF of two numbers can be found using various methods, including the prime factorization method, the Euclidean algorithm, and the Venn diagram method.
• The LCM and HCF are important concepts in number theory and have applications in various mathematical problems, such as simplifying fractions, solving equations, and finding the least common denominator.