Question

What is the least common multiple of $36$ and $12$ ?

Answer 1

Hint: The least common multiple (LCM) can be found by using the prime factorization method, division method or, listing all the multiples and picking the smallest common multiple. Here we are asked to find the least common multiple of $36$ and $12$. We shall apply the third method, writing all the multiples and obtaining the smallest common multiple. This method is the easiest of the three methods.

Complete step-by-step answer:
First, we shall list all the multiples of the given numbers. Then we need to pick the smallest common multiple which is the required LCM of given numbers.
We shall note that the term multiple refers to the multiples of the given number whereas the term common multiple is to list the multiples that are common for the given numbers.
Here, we are given $36$ and $12$.
We shall list out the multiples of $36$ and $12$.
The multiples of $36$ are $36,72,108,..$
The multiples of $12$are $12,24,36,48,60,72,84,...$
Now, we need to pick the common multiple.
Thus, $36$ is the common multiple of $36$ and $12$.
Hence $36$ is the least common multiple of $36$ and $12$.

Note: Now, we shall use the prime factorization method to verify the obtained answer. We need to calculate the prime factors of the given numbers and we need to multiply the different factors that have the highest exponent.
The prime factors of $36$ are $2 \times 2 \times 3 = {2^2} \times 3$
The prime factors of $12$are $2 \times 2 \times 3 \times 3 = {2^2} \times {3^2}$
Here, we can note that the different factors are $2$ and $3$.
We need to pick the factors $2$ and $3$ having the highest exponent.
Thus, we have ${2^2} \times {3^2}$
${2^2} \times {3^2} = 36$
Hence $36$ is the least common multiple of $36$ and $12$.
Therefore, the obtained answer is verified.