Question

What is the least common multiple of 8,12 and 5?

Answer 1

Hint: The LCM which is a short form for Least Common Multiple is the least common multiple of two or more than two numbers. Unlike the factors of a number, which is always a prime number, it can take any value. We will write the multiples of the given numbers and then find the smallest number which is common for all. This can be done by using prime factorization.

Complete step-by-step solution:
We have the given set of numbers as: 8, 12 and 5.
Now, we will calculate the least common multiple of these numbers by the method of prime factorization.
For that, we will first break these numbers as their prime factors and then list down the prime numbers, as many number of times, as in the terms in which they occur the most. The product of these terms will give us the required LCM.
On writing the terms as a product of common prime factors, we get:
$\begin{align}
  & \Rightarrow 8=2\times 2\times 2 \\
 & \Rightarrow 12=2\times 2\times 3 \\
 & \Rightarrow 5=5 \\
\end{align}$
Now, taking the prime terms with highest repetition, we get:
$=(2\times 2\times 2),(3)\text{ and (5)}$
The product of these terms is equal to:
$\begin{align}
  & =2\times 2\times 2\times 3\times 5 \\
 & =120 \\
\end{align}$
Hence, the least common multiple or LCM of 8,12 and 5 comes out to be 120.

Note: One should never misinterpret the term LCM with HCF as both of them have totally opposite meanings. HCF of a set of numbers is the highest common factor of the set of numbers. We can remember the difference as, HCF of a set of numbers is always less than the smallest number and LCM of a set of numbers is always greater than the largest number.