Finding the LCM by Prime factorization Method

The prime factors method is another most useful mathematical approach for finding the least common multiple of two or more quantities. In this method, a lowest common multiple is calculated by splitting each of the given number as prime factors, and then evaluated the product of the highest power prime factors of all given numbers.

Procedure

In prime factoring method, there are three simple steps involved to find the least common multiple of the given two or more numbers.

Express each number as a product of prime factors.
Write Prime factors in Index form by exponentiation.
Find the product of all highest power prime factors of the given numbers.

Example

$8$, $12$ and $24$ are three given numbers.

Firstly, factorize each quantity as prime factors by using factorization and then write prime factors in exponential notation by the exponentiation.

$(1) \,\,\,$ $8 \,=\, 2 \times 2 \times 2$
$\,\,\,\,\,\,\,\,\,\, \implies$ $8 \,=\, 2^3$

$(2) \,\,\,$ $12 \,=\, 2 \times 2 \times 3$
$\,\,\,\,\,\,\,\,\,\, \implies$ $12 \,=\, 2^2 \times 3^1$

$(3) \,\,\,$ $24 \,=\, 2 \times 2 \times 2 \times 3$
$\,\,\,\,\,\,\,\,\,\, \implies$ $12 \,=\, 2^3 \times 3^1$

Observe the three cases and identify the highest power prime factors. $2^3$ and $3^1$ are the highest power prime factors in exponential form. Now, find the product of them to calculate the least common multiple by the prime factorisation method.

$LCM$ $\,=\,$ $2^3 \times 3^1$

$\implies$ $LCM$ $\,=\,$ $8 \times 3$

$\,\,\, \therefore \,\,\,\,\,\,$ $LCM$ $\,=\,$ $12$