The least common multiple or lowest common multiple (LCM) or smallest common multiple of two integers \(a\) and \(b\) is the smallest positive integer that is a multiple of both \(a\) and \(b\). Since it is a multiple, it can be divided by \(a\) and \(b\) without a remainder. For example, the least common multiple of the numbers 14 and 6 is 42.

The unique factorization theorem says that every positive integer number greater than 1 can be written in only one way as a product of prime numbers. The prime numbers when combined together, make up a composite number.

Example : Find the value of  LCM (8,9,21).

Solution: First, factor out each number and express it as a product of prime number powers. The LCM will be the product of multiplying the highest power in each prime factor category together. Out of the 4 prime factor categories 2, 3, 5, and 7, the highest powers from each are \({2^3},{\rm{ }}{3^2},{\rm{ }}{5^0},\) and \({7^1}\). Thus, LCM = 23.32.7 = 504.