Factorisation of 108: A Complete Guide to Breaking Down a Key Number

Understanding the factorisation of a number is a fundamental concept in mathematics that reveals the building blocks of that number. Whether you're solving equations, simplifying fractions, or exploring prime numbers, factorisation plays a crucial role. In this article, we’ll dive into the factorisation of 108, exploring how to break it down into prime factors and beyond. This guide is tailored for students, educators, and math enthusiasts wanting a clear, detailed look at one of the most commonly studied numbers in arithmetic.

Understanding the Context

What is Factorisation?

Factorisation is the process of expressing a number as a product of its prime or composite factors. When we talk about the factorisation of 108, we’re identifying which prime numbers multiply together to give 108.

Prime Factorisation of 108

factorisation of 108

Key Insights

To fully understand 108, we perform prime factorisation — breaking it down into prime numbers only.

Step-by-step Prime Factorisation:

  1. Start with the smallest prime number (2):
    108 is even, so divide by 2 →
    $ 108 ÷ 2 = 54 $

  2. Continue dividing by 2:
    $ 54 ÷ 2 = 27 $
    So far: $ 108 = 2 × 2 × 27 = 2² × 27 $

  3. Now work with 27, which is not divisible by 2, move to next prime: 3
    $ 27 ÷ 3 = 9 $

Continue Reading

Let $ A = x^2 + 4 $, $ B = y^2 + 2 $, and $ C = y^2 - 2 $, then:
A^2 - 16BC = (A - 4\sqrt{BC})(A + 4\sqrt{BC})
But this factoring is not ideal over real numbers. Instead, observe that $ x^4 - 16y^4 = (x^2)^2 - (4y^2)^2 = (x^2 - 4y^2)(x^2 + 4y^2) $.

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Final Thoughts

  1. Again divide by 3:
    $ 9 ÷ 3 = 3 $

  2. Finally:
    $ 3 ÷ 3 = 1 $

Final Prime Factorisation:

Putting it all together, we get:

$$
108 = 2^2 × 3^3
$$

This means 108 is the product of $2^2$ (two twos) and $3^3$ (three threes).

Why Factorise 108? – Key Benefits

  1. Simplifying Fractions
    Knowing that $108 = 2^2 × 3^3$ helps simplify fractions efficiently, especially when dealing with LCMs and GCFs.

  2. Finding LCMs and GCFs
    Factorisation allows quick computation of least common multiples and greatest common factors — essential in algebra, number theory, and real-world problem solving.