Question

How do you find the prime factorization of $54$?

Answer 1

Hint: In this we will proceed by dividing $54$ by the smallest prime number that divides it completely. The quotient obtained will again be divided by the smallest prime number that divides it completely. Hence we have to do this till we get the quotient as $1$ and then all the divisors are multiplied to get the prime factorization of that number $54$.

Complete step-by-step answer:
Here we are given to find the prime factorization of the number $54$
So we must know what the prime factorization means. It is actually finding the number in the form of multiplication of all the prime numbers that divide that number completely. The example will make it clearer.
For example: If we have the number $20$ and we need to find its prime factorization, we will firstly find the smallest prime number that divides it completely. We know that it is $2$ now we get $\dfrac{{20}}{2} = 10$ and now we will see the result that is $10$
It is again divisible by the smallest prime number $2$ so we get $\dfrac{{10}}{2} = 5$
Now we are left with $5$ now the smallest prime number that divides it is $5$ so we get $\dfrac{5}{5} = 1$
So we can write all the divisors’ multiplication that is $20 = 2 \times 2 \times 5 = {2^2} \times 5$ as its prime factorization.
Now we will see the problem where we are given the number $54$
So we have the smallest prime number $2$ that divides it completely. Hence we will divide and get:
$\dfrac{{54}}{2} = 27$
Now we get $27$ so we will see which prime number divides it completely so that is $3$ and we get:
$\dfrac{{27}}{3} = 9$
Now we get $9$ as the result and again we have smallest prime number $3$ that divides it completely so we get:
$\dfrac{9}{3} = 3$
Now we get $3$ and it is divisible by $3$ itself and we get:
$\dfrac{3}{3} = 1$
Hence we have reached the position where we have got the quotient as $1$
So we can multiply all the divisors and get the prime factorization of the number $54$ as $2 \times 3 \times 3 \times 3$
Hence we can write it as $2 \times {3^3}$.

Note: In these types of problems we just need to make the division reach till the quotient becomes $1$ and another thing the student must keep in mind is that only the prime numbers can be the divisors while finding the prime factorization of any number.