Introduction:

HCF and LCM are two important mathematical concepts that are commonly tested in competitive exams. HCF stands for Highest Common Factor and LCM stands for Least Common Multiple. Both concepts are used to find the factors and multiples of numbers, and are essential for solving mathematical problems related to fractions, ratios, and proportions.

What is HCF?

HCF is the greatest common factor of two or more numbers. It is the largest number that divides all the numbers exactly without leaving a remainder. For example, the HCF of 12 and 16 is 4, as 4 is the largest number that divides both 12 and 16 exactly. To find the HCF of two or more numbers, you can use the Euclidean algorithm or the prime factorization method.

Euclidean Algorithm

The Euclidean algorithm is a step-by-step method for finding the HCF of two numbers. It involves dividing the larger number by the smaller number and noting the remainder. Then, the smaller number is divided by the remainder, and the process is repeated until the remainder is zero. The last non-zero remainder is the HCF of the two numbers.

Prime Factorization Method

The prime factorization method involves finding the prime factors of each number and then finding the common prime factors. For example, to find the HCF of 12 and 16, we first find the prime factors of 12 (2x2x3) and the prime factors of 16 (2x2x2x2). The common prime factors are 2x2, so the HCF is 2x2 = 4.

What is LCM?

LCM is the least common multiple of two or more numbers. It is the smallest number that is a multiple of all the numbers. For example, the LCM of 12 and 16 is 48, as 48 is the smallest number that is a multiple of both 12 and 16. To find the LCM of two or more numbers, you can use the prime factorization method or the list method.

Prime Factorization Method

The prime factorization method for finding LCM is similar to the method for finding HCF. The prime factors of each number are found and then the highest power of each prime factor is used to form the LCM. For example, to find the LCM of 12 and 16, we first find the prime factors of 12 (2x2x3) and the prime factors of 16 (2x2x2x2). The LCM is formed by multiplying the highest power of each prime factor, which is 2x2x3x2 = 48.

List Method

The list method for finding LCM involves listing the multiples of each number and finding the smallest multiple that is common to all numbers. For example, to find the LCM of 12 and 16, we list the multiples of 12 (12, 24, 36, 48, 60, 72, etc.) and the multiples of 16 (16, 32, 48, 64, 80, 96, etc.). The smallest multiple that is common to both lists is 48.

In conclusion, LCM and HCF are important mathematical concepts that are commonly tested in competitive exams. Understanding the concepts and the different methods for finding them can help you to solve mathematical problems related to fractions, ratios, and proportions. It's recommended to practice with a lot of problems to feel comfortable with these concepts during a test.

HCF & LCM