Question

How do I express $108$ as a product of prime factors?

Answer 1

Hint: To express $108$ as a product of prime numbers, we should carry out prime factorisation of it. We should start dividing it by $2$ as it is the smallest prime number and then by $3,5$ and others as needed.

Complete step by step solution:
The given number that we have at our disposal is
$108$
Every integer can be either prime or composite. The definition of a prime number is that a prime number is a number which is not divisible by any number other than $1$ and the number itself. Examples are $2,3,5$ . The definition of a composite number is that a composite number is a number which is divisible by at least one number other than $1$ and the number itself. Examples are $4,6,8$ which are divisible by $2$ also. Prime numbers are the building blocks of composite numbers.
Now, the given number being $108$ , we can see that the unit’s place digit is $8$ . This means that $108$ is clearly an even number. All even numbers except $2$ are composite numbers, so $108$ is a composite number.
$108$ upon division by $2$ gives $\dfrac{108}{2}=54$ . $54$ upon division by $2$ gives $27$ . $27$ upon division by $3$ gives $9$ . $9$ upon division by $3$ gives $3$ itself. $3$ being a prime number itself, cannot be further prime factored. Thus the entire prime factorisation can be written as,
$108=2\times 2\times 3\times 3\times 3$
Since here are two $2s$ and three $3s$ , we can rewrite it as,
$\Rightarrow 108={{2}^{2}}\times {{3}^{3}}$
Therefore, we can conclude that $108$ can be expressed as a product of prime factors by ${{2}^{2}}\times {{3}^{3}}$.

Note: While dividing a composite number by a prime number repeatedly, we should carefully count the number of divisions for each prime number otherwise, any mistake will give a wrong answer. We should remember to express the number of divisions of each prime number as a power to it. Also, at the end we should cross check the product of the prime numbers.