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How do you find the least common multiple of $28,\;14,\;21$?

Last updated date: 13th Jun 2024
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Hint: In this question, we find the least common multiple of three numbers. We find the least common multiple by two methods. The first method is, we find the multiple of given numbers. Then we choose the common number in the multiple of these numbers, the common number is the least common multiple of the given three numbers. And the second method is, we find the factor of each number. Then we check each number from the factorization of each given number, which comes maximum time. After that multiply, these numbers and the number which comes after multiplication is the least common multiple.

Complete step by step solution:
In this question, the word least common multiple is used. First, we know about the least common multiple. The least common multiple is defined as the smallest common multiple of two or more numbers.
It is denoted as L.C.M.
Now we come to the question. To find the least common multiple of three numbers, there are two methods.
First method:
First, we find the multiple of given numbers $\left( {28,\;14,\;21} \right)$ as below. And choose the non-zero common multiple.
Now,
We find the multiple of $28$.
The multiple of $28$ is as below.
$28 = 0,\;28,\;56,\;84.........\;\;\left( 1 \right)$
Then,
We find the multiple of $14$.
The multiple of $14$ is as below.
$14 = 0,14,28,42,56,70,84.........\left( 2 \right)$
After that,
We find the multiple of $21$.
The multiple of $21$ is as below.
$\Rightarrow 21 = 0,21,42,63,84.........\left( 3 \right)$
Now,
We find the common form of the equation $\left( 1 \right),\;\left( 2 \right)$ and $\left( 3 \right)$.
Thus, the common number in the factors of $28,14,21$ is $84$.
Therefore, the least common multiple of $\left( {28,\;14,\;21} \right)$ is $84$.
Second method:
We write the prime factorization of $28,14,21$
In prime factorization, we take all the prime numbers with the highest exponent.
Then,
The prime factors of $28$ are.
$\Rightarrow 28 = 4 \times 7 = {2^2} \times 7$
The prime factors of $14$ are.
$14 = 2 \times 7$
And the prime factors of $21$ are.
$21 = 3 \times 7$
Then we build a table of prime factors. And the table is given below.
 Prime factor Number $14$ Number $21$ Number $28$ L.C.M (max) $2$ $1$ $0$ $2$ $2$ $3$ $0$ $\;1$ $0$ $1$ $7$ $1$ $1$ $1$ $1$

Therefore, the least common multiple of $28,14$ and $21$ is as below.

$\therefore L.C.M\left( {28,14,21} \right) = {2^2} \times 3 \times 7 = 84$

Note:
If you have three numbers and you want to find the least common multiple of these three numbers. Then first you find the multiple of the given three numbers. After that, you find the common number from multiple of given numbers. Thus, the common number from multiple of given numbers is the least common multiple.