How to find the Least or Lowest Common Multiple by the Prime factors Method

Introduction

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.

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

There are three simple steps to find the least common multiple of two or more real numbers.

1. Express each given number as the product of prime numbers.
2. According to the exponentiation, write the prime factors of every number in exponent or index form.
3. 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.

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

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

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

Worksheet

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