Question

Find the prime factorization of 54.

Answer 1

Hint: Consider one of the factorizations of 54. Express each factor in it in the form of product
of their prime factors. We do this until each number in the factorization is a prime number.
The expression thus obtained is the correct answer.

Complete step by step solution:
Prime factorization is the process of splitting the number into a product of its prime factors.
We know that there are various ways in which 54 can be written as a product of some numbers.
For example, consider the following factorizations of 54.
$$\begin{gathered} 54=2\times 27 \\ 54=3\times 18 \\ 54=6\times 9 \end{gathered}$$
The reason the above expressions are not considered as an answer to our question is that the factors
given are not all primes.
So, what we are looking for is the ‘prime’ factorization.
The word ‘prime’ is the key here. It is a must that every number in the factorization is a prime number.
While there can be many factorizations of a number, there can only be one prime factorization.
We know that the factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54.
We will use the smallest prime factor of 54 which is 2 to divide 54.

$\Rightarrow 54=2\times 27$
But 27 is not a prime number as it is divisible by 3.
So, we further split 27 into its factors.
The smallest prime factor for 27 is 3.
Therefore, we have $54=2\times 3\times 9$
Since 9 is not a prime number, we split it further.
Here is what we get-
$54=2\times 3\times 3\times 3$
Now, this factorization contains only primes and hence it is called the prime factorization.
Hence the required answer is $2\times 3\times 3\times 3$or $2\times 3^3$

Note: It is not necessary to begin the factorization using the smallest prime factor.
So, we may also do it the following way:
$54=6\times 9$
Here neither 6 nor 9 is a prime number. So, we need to factorize both of them.
$$\begin{gathered} 54=6\times 9 \\ =2\times 3\times 3\times 3 \end{gathered}$$
We can see that this method also gives us the same answer.