Question

What is the least common multiple of 36 and 54?

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 solution:
Here we have been asked to find the least common multiple of the numbers: 36 and 54. First, let us know about L.C.M.
In arithmetic and number theory, the least common multiple (L.C.M) also called the first common multiple of two or more integers is the smallest positive integer that is divisible by each of the given integers. 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. So writing the numbers 36 and 54 as the product of their prime factors we get,
\[\Rightarrow 36=2\times 2\times 3\times 3={{2}^{2}}\times {{3}^{2}}\]
\[\Rightarrow 54=2\times 3\times 3\times 3=2\times {{3}^{3}}\]
Clearly we can see that the highest power of the prime factors 2 is 2 and 3 is 3. So, we need to multiply ${{2}^{2}}$ and ${{3}^{3}}$ to get the L.C.M.
\[\Rightarrow \] L.C.M = \[{{2}^{2}}\times {{3}^{3}}\]
\[\Rightarrow \] L.C.M = \[4\times 27\]
$\therefore $ L.C.M = 108
Hence the L.C.M of 36 and 54 is 108.

Note: Here we can also use a different approach to get the L.C.M of the given numbers. What we will do is we will write a few multiples of 36 and 54 and check which multiple will appear first and is common in both. However the problem arises due to the fact that the numbers are large and we initially don’t know how many multiples we need to write for both.