Question

What is the least common multiple of 12 and 16?

Answer 1

Hint: To find the L.C.M first we will write the given numbers as the product of their prime factors. Now, if any factor will be repeating then we will write them in exponential form. Finally, we will take the prime factors with their highest exponent present and multiply them together to get the required L.C.M of the given numbers.

Complete step by step answer:
Here, we have been asked to find the L.C.M of these given numbers: 12 and 16. First, let us know about L.C.M.
In arithmetic and number theory, the least common multiple of two integers x and y is the smallest positive integer that is divisible by both x and y. Here, x and y must not be 0. There are many 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 power present.
Now let us come to the question. Here we have two numbers 12 and 16, so writing them as the product of their primes we get,
\[\begin{align}
  & \Rightarrow 12=2\times 2\times 3={{2}^{2}}\times {{3}^{1}} \\
 & \Rightarrow 16=2\times 2\times 2\times 2={{2}^{4}} \\
\end{align}\]
Clearly we can see that the highest power of the prime factors 2 is 4 and 3 is 1. So, we need to multiply ${{2}^{4}}$ with ${{3}^{1}}$ to get the L.C.M.
\[\Rightarrow \] L.C.M = \[{{2}^{4}}\times {{3}^{1}}\]
\[\Rightarrow \] L.C.M = \[16\times 3\]
$\therefore $ L.C.M = 48

Hence, the L.C.M of 12 and 16 is 48.

Note: We can also find the L.C.M by writing the multiples of the given numbers and then check the first common multiple that will appear like the multiples of 12 are: 12, 24, 36, 48, 60,…… and the multiples of 16 are: 16, 32, 48, 64……. So clearly we can see that the first common multiple is 48 which will be our answer. But remember that this process is only beneficial when the numbers are small. In addition to L.C.M remember that in the process of finding H.C.F we consider only common prime factors that are present in the prime factorization of given numbers.