HCF stands for Highest Common Factor, and LCM stands for Least Common Multiple. HCF is the largest number that divides two or more given numbers without leaving a remainder, while LCM is the smallest number that is a multiple of two or more given numbers without leaving a remainder.

To find the HCF and LCM, there are two important methods: the prime factorization method and the division method. The relationship between HCF and LCM is that the product of HCF and LCM is equal to the product of the given numbers.

Here are some examples to illustrate the concepts:

• HCF of 8 and 12: The factors of 8 are 1, 2, 4, and 8. The factors of 12 are 1, 2, 3, 4, 6, and 12. The highest number that can divide both 8 and 12 is 4, so the HCF of 8 and 12 is 4.

• LCM of 8 and 12: The multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, and so on. The multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, and so on. The smallest number that appears in both lists is 24, so the LCM of 8 and 12 is 24.

• HCF and LCM of 60 and 72: The prime factors of 60 are 2, 2, 3, and 5. The prime factors of 72 are 2, 2, 2, 3, and 3. The highest common factor is found by multiplying all the factors which appear in both lists, which is 2 x 2 x 3 = 12. The lowest common multiple is found by multiplying all the factors which appear in either list, which is 2 x 2 x 2 x 3 x 3 x 5 = 360.

There are also shortcut methods to find HCF and LCM, such as using the division method.