Question

What is the least common multiple of 16, 18 and 9?

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: 16, 18 and 9. First, let us know about the term ‘least common multiple’.
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 numbers. 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. Writing the given numbers as the product of their prime factors we get,
$\Rightarrow 16=2\times 2\times 2\times 2={{2}^{4}}$
\[\Rightarrow 18=2\times 3\times 3=2\times {{3}^{2}}\]
\[\Rightarrow 9=3\times 3={{3}^{2}}\]
Clearly we can see that the highest power of the prime factors 2 is 4 and 3 is 2. So we need to multiply \[{{2}^{4}}\] and ${{3}^{2}}$ to get the L.C.M.
\[\Rightarrow \] L.C.M = \[{{2}^{4}}\times {{3}^{2}}\]
\[\Rightarrow \] L.C.M = \[16\times 9\]
$\therefore $ L.C.M = 144
Hence, the L.C.M of 16, 18 and 9 is 144.

Note: Note that the other method by which we can find the L.C.M is that we will write the multiples of 16, 18 and 9 one – by – one and check which multiple occurs first and is common in all of them. But this method will be preferred when we have to find the L.C.M of two numbers and they are small in magnitude because initially we don’t know how many multiples we need to write. In case the numbers provided are large then prime factorization is the best approach.