Question

What is the least common multiple of 3 and 4?

Answer 1

Hint: We solve this problem by using the prime factorisation method. The prime factorisation method is the method used to represent any number as the product of prime numbers.
We represent the numbers 3 and 4 as products of prime numbers then we take the LCM as the product of common primes in both numbers along with the remaining primes after taking out the common primes. For example, if the numbers are in form,
$\begin{align}
  & X={{2}^{2}}\times 3\times {{5}^{3}} \\
 & Y=2\times {{5}^{2}}\times 7 \\
\end{align}$
Then the LCM of $X,Y$ is given as,
$\Rightarrow LCM\left( X,Y \right)=\left( 2\times {{5}^{2}} \right)\times \left( 2\times 3\times 5\times 7 \right)$

Complete step-by-step solution:
We are asked to find the least common multiple of 3 and 4.
Let us use the prime factorisation method for solving this problem.
We know that the method of representing the numbers as the product of prime numbers is called the prime factorisation method.
Now, let us represent the number 3 as product of prime numbers then we get,
$\Rightarrow 3=3$
Now, let us represent the number 4 as the product of prime numbers then we get,
$\Rightarrow 4={{2}^{2}}$
Let us assume that the required LCM of 3 and 4 as $'L'$
We know that the LCM of two numbers from the prime factorisation method is calculated as a product of common primes in both numbers along with the remaining primes after taking out the common primes.
Now, by using the above theorem we get the required LCM of numbers 3 and 4 as,
$\begin{align}
  & \Rightarrow L=3\times {{2}^{2}} \\
 & \Rightarrow L=12 \\
\end{align}$
Therefore, we can conclude that the required LCM of given numbers 3 and 4 is 12 that is,
$\therefore LCM\left( 3,4 \right)=12$

Note: We can solve this problem using the division method also.
Let us take the numbers 3 and 4 and divide them using the first prime number that is 2 and taking the quotient in the next line then we get,
\[\begin{align}
  & \underline{\left. 2 \right|3\text{ }4} \\
 & \underline{\left. \text{ } \right|3\text{ 2}} \\
\end{align}\]
Now, let us divide the above numbers again until we get the quotients as 1 then we get,
\[\begin{align}
  & \underline{\left. 2 \right|3\text{ }4} \\
 & \underline{\left. 2 \right|3\text{ 2}} \\
 & \underline{\left. 3 \right|3\text{ }1} \\
 & \underline{\left. \text{ } \right|1\text{ }1} \\
\end{align}\]
Now, we know that the product of all prime numbers that are used in division gives the LCM of the numbers.
By using this result we get the required LCM of 3 and 4 as,
$\begin{align}
  & \Rightarrow LCM\left( 3,4 \right)=2\times 2\times 3 \\
 & \Rightarrow LCM\left( 3,4 \right)=12 \\
\end{align}$
Therefore, we can conclude that the required LCM of given numbers 3 and 4 is 12 that is,
$\therefore LCM\left( 3,4 \right)=12$