Key Points

The Highest Common Factor (HCF) of two or more numbers is the largest number that divides each of them without leaving a remainder. It is also known as the Greatest Common Divisor (GCD).

The Lowest Common Multiple (LCM) of two or more numbers is the smallest positive number that is a multiple of all the given numbers.

Prime factorization is the process of expressing a composite number as a product of its prime factors. For example, the prime factorization of $84$ is $2 imes 2 imes 3 imes 7$.

To find the HCF, multiply the common prime factors, each raised to its lowest power found in the numbers. For example, if $A = 2^3 imes 3^1$ and $B = 2^2 imes 3^2$, then $ext{HCF}(A, B) = 2^2 imes 3^1 = 12$.

To find the LCM, multiply all the prime factors (common and uncommon) from the numbers, each raised to its highest power. For example, if $A = 2^3 imes 3^1$ and $B = 2^2 imes 3^2 imes 5$, then $ext{LCM}(A, B) = 2^3 imes 3^2 imes 5^1 = 360$.

For any two positive integers $a$ and $b$, the product of the numbers is equal to the product of their HCF and LCM. The formula is $ext{HCF}(a, b) imes ext{LCM}(a, b) = a imes b$.

A shortcut to find HCF and LCM is to divide the numbers by common factors. The HCF is the product of the common divisors, while the LCM is the product of all divisors and the remaining undivided numbers.

Two numbers are co-prime if their only common factor is 1. For co-prime numbers, the HCF is always 1 and the LCM is their product.

The HCF of any two consecutive numbers is always 1. For example, $ext{HCF}(20, 21) = 1$. The HCF of two consecutive even numbers is always 2.

If a number $a$ is a factor of another number $b$, then their HCF is the smaller number $a$, and their LCM is the larger number $b$. For example, for 7 and 21, $ext{HCF}(7, 21) = 7$ and $ext{LCM}(7, 21) = 21$.

HCF is used in problems to find the largest size of an item that can be used to measure or divide quantities exactly. For example, finding the largest square tile to pave a rectangular floor.

LCM is used to find the time or point at which events with different cycles will occur together. For example, finding when two bells ringing at different intervals will ring at the same time.