**Highest Common Factor (HCF)**

The Highest Common Factor (HCF) of two or more numbers is the largest number that divides each of the given numbers without leaving a remainder. It is also known as the Greatest Common Divisor (GCD). Finding the HCF of two numbers helps in simplifying fractions and solving various mathematical problems.

To determine the HCF of two or more numbers, the following steps can be followed:

1. **Prime Factorization Method**: First, find the prime factors of each number. Prime factors are the prime numbers that can divide the given number without leaving a remainder. Express each number as a product of its prime factors.

2. **Identify Common Prime Factors**: Compare the prime factors of all the numbers. Identify the common prime factors among them. These are the prime factors that are present in all the given numbers.

3. **Multiply Common Prime Factors**: Multiply all the common prime factors. The product obtained is the HCF of the given numbers.

For example, let's find the HCF of 24 and 36 using the prime factorization method:

- Prime factorization of 24: 2 × 2 × 2 × 3
- Prime factorization of 36: 2 × 2 × 3 × 3

The common prime factors are 2 and 3. Multiplying them together gives us 2 × 2 × 3 = 12. Therefore, the HCF of 24 and 36 is 12.

**Lowest Common Multiple (LCM)**

The Lowest Common Multiple (LCM) of two or more numbers is the smallest number that is divisible by each of the given numbers. It is helpful in solving problems related to time, distance, and finding a common denominator for fractions.

To find the LCM of two or more numbers, the following steps can be followed:

1. **List the Multiples**: Write down the multiples of each number until you find a common multiple. A multiple is obtained by multiplying a number by any whole number.

2. **Identify the Common Multiple**: Identify the smallest common multiple among the listed multiples. This is the LCM of the given numbers.

For example, let's find the LCM of 6 and 8:

- Multiples of 6: 6, 12, 18, 24, 30, 36, ...
- Multiples of 8: 8, 16, 24, 32, 40, 48, ...

The common multiple is 24. Therefore, the LCM of 6 and 8 is 24.

In conclusion, the HCF is the largest number that divides all the given numbers without leaving a remainder, while the LCM is the smallest number that is divisible by all the given numbers. These concepts are widely used in mathematics to simplify fractions, solve equations, and find common denominators.