Question

What is the least common multiple of 3, 4 and 5?

Answer 1

Hint: Here we will use the prime factorization method to find the L.C.M. First we will write the given numbers as the product of their prime factors one – by – one. Now, if a prime factor will be repeating then we will write them in exponential form. Finally, we will take the product of all the different prime factors along with their highest exponent to get the answer.

Complete step by step answer:
Here, we have been asked to find the L.C.M of three numbers: 3, 4 and 5. First, let us know about L.C.M.
In arithmetic and number theory, the least common multiple of two or more integers is the smallest positive integer that is divisible by each of the given. The given integers must not be 0. There are two methods to determine the L.C.M of two or more given numbers. Here, we will use the method of prime factorization.
In the method of prime factorization we write the given numbers as the product of their prime factors. Now, the L.C.M will be the product of the prime factors along with their highest exponent present.
Now let us come to the question. Here we have three numbers: 3, 4 and 5. Since 3 and 5 are already prime numbers so they cannot be factored further but we can factor 4, so we get,
\[\begin{align}
  & \Rightarrow 3={{3}^{1}} \\
 & \Rightarrow 4=2\times 2={{2}^{2}} \\
 & \Rightarrow 5={{5}^{1}} \\
\end{align}\]
Clearly we can see that the highest power of the prime factors 2 is 2, 3 is 1 and 5 is 1. So, we need to multiply ${{2}^{2}}$, ${{3}^{1}}$ and ${{5}^{1}}$ to get the L.C.M.
\[\Rightarrow \] L.C.M = \[{{2}^{2}}\times {{3}^{1}}\times {{5}^{1}}\]
\[\Rightarrow \] L.C.M = \[4\times 3\times 5\]
$\therefore $ L.C.M = 60

Hence the L.C.M of 3, 4 and 5 is 60.

Note: There is one more method by which we can find the L.C.M. In that method we will write the multiples of 3, 4 and 5 one – by – one and check which multiple occurs first. But this method may not be preferred more because initially we don’t know how many multiples we need to write. In case the numbers provided are large then prime factorization is the best approach. Also, you may remember a short way to find the L.C.M of two prime numbers which is: ‘the L.C.M of two prime numbers is their product’. This is because they cannot be factored.