Question

Find the LCM of $24$, $36$, and $48$

Answer 1

Hint: There are various methods for finding the least common multiple of three given numbers. The simplest method to find LCM is by prime factorization method. Least common multiple is a product of common factors with highest power and all other non-common factors. So, we will find the prime factors of all the three numbers first and then find LCM by multiplying the common factors with highest powers and the non-common factors.

Complete step-by-step answer:
The numbers given to us in the question are: $24$, $36$, and $48$.
We have to find the least common multiple of the given three numbers by prime factorization method.
In prime factorization, we represent the numbers as a product of their constituent prime factors. So, we get,
Prime factors of $24$$ = 2 \times 2 \times 2 \times 3$
$ = {2^3} \times 3$
Prime factors of $36$$ = 2 \times 2 \times 3 \times 3$
$ = {2^2} \times {3^2}$
Prime factors of $48$$ = 2 \times 2 \times 2 \times 2 \times 3$
$ = {2^4} \times 3$
Now, Least common multiple is a product of common factors with highest power and all other non-common factors. So, finding the least common multiple of the three numbers, we get,
Least common multiple of $24$, $36$ and $48$$ = {2^4} \times {3^2}$
$ = 16 \times 9$
$ = 144$
Hence, the least common multiple of $24$, $36$ and $48$ is $144$.
So, the correct answer is “144”.

Note: Knowledge of least common multiple is also used in addition and subtraction of fractions. To find the LCM of numbers, we first express them in product form of prime numbers. It is important to note here that a composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than $1$ and itself. The prime factorization of the prime number is the number itself and $1$.