Prime Factorization Examples for Beginners (With Steps)

The fastest way to master prime factorization is to work through plenty of examples. This reference article factorises the numbers that appear most often in Indian school textbooks and exams, showing each step so you can follow the pattern and apply it to any number. You can reproduce or extend any of these with a free prime factorization calculator.

Key takeaway: Once you can factorise the common numbers up to 100 quickly, larger numbers become easy because you simply keep applying the same small primes.

Example 1: Prime Factorization of 24

Divide 24 by 2 to get 12, by 2 again to get 6, by 2 again to get 3, and 3 is prime. So 24 = 2 × 2 × 2 × 3, which we write neatly as 2³ × 3. This is a very common number in exams, so it is worth remembering.

Example 2: Prime Factorization of 36

Divide 36 by 2 to get 18, by 2 to get 9, then 9 by 3 to get 3, and 3 by 3 to reach 1. So 36 = 2 × 2 × 3 × 3, or 2² × 3². Notice that 36 is a perfect square, and its prime factorization has only even exponents, which is true of every perfect square.

Example 3: Prime Factorization of 45

45 is odd, so 2 does not divide it. Try 3: 45 divided by 3 is 15, and 15 divided by 3 is 5, which is prime. So 45 = 3 × 3 × 5, or 3² × 5. This example shows why you move to the next prime when the smallest one does not fit.

Example 4: Prime Factorization of 84

Divide 84 by 2 to get 42, by 2 to get 21, then 21 by 3 to get 7, which is prime. So 84 = 2 × 2 × 3 × 7, or 2² × 3 × 7. This number has three different primes, a good reminder that factorisations can mix several primes together.

Example 5: Prime Factorization of 100

Divide 100 by 2 to get 50, by 2 to get 25, then 25 by 5 to get 5, and 5 by 5 to reach 1. So 100 = 2 × 2 × 5 × 5, or 2² × 5². Like 36, it is a perfect square with even exponents.

Reference Table of Common Numbers

Using the Examples for HCF and LCM

Once numbers are in prime form, HCF and LCM become easy. Take 60 = 2² × 3 × 5 and 72 = 2³ × 3². The HCF uses the lowest power of each shared prime, giving 2² × 3 = 12. The LCM uses the highest power of every prime, giving 2³ × 3² × 5 = 360. You can confirm the arithmetic with a multiplication calculator and check that the HCF times the LCM equals the product of the two numbers.

Benefits of Practising With Examples

Working through many examples builds the speed and confidence that exams reward. You start to recognise common factorisations at a glance, spot perfect squares by their even exponents, and choose the right starting prime instantly. This fluency carries directly into fraction simplification, HCF and LCM problems, and the number-theory questions that appear in competitive exams.

Challenges and Limitations

These examples use numbers up to a few hundred for clarity, but very large numbers take longer and are better handled by a calculator. Students sometimes memorise answers without understanding the steps, which fails them when a new number appears. The method also applies only to whole numbers, so it does not directly cover fractions or decimals.

Common Mistakes to Avoid

• Trying 2 on an odd number. If the number is odd, skip straight to 3.
• Stopping before reaching 1. Keep dividing until nothing but primes remain.
• Missing a repeated prime. Divide by the same prime as many times as it fits.
• Writing the wrong exponent. Count carefully how many times each prime is used.
• Forgetting to check. Multiply the primes back to confirm the original number.
• Memorising instead of understanding. Learn the steps, not just the answers.

Best Practices and Expert Recommendations

• Practise a mix of numbers. Include odd, even, and perfect-square numbers.
• Use divisibility rules. They tell you which prime to try first.
• Write answers in exponent form. It is neat and expected in exams.
• Link to HCF and LCM. Apply your factorisations to two-number problems.
• Verify each answer. Multiply the primes back to be sure.
• Build up gradually. Master small numbers before moving to large ones.

Example 6: A Three-Digit Number, 540

Larger numbers use exactly the same steps. Take 540. Divide by 2 to get 270, by 2 to get 135. Now 135 is odd, so move to 3: 135 divided by 3 is 45, 45 divided by 3 is 15, and 15 divided by 3 is 5, which is prime. So 540 = 2 × 2 × 3 × 3 × 3 × 5, or 2² × 3³ × 5. The only difference from the smaller examples is that you divide a few more times, which shows that no number is really harder, just longer.

Spotting Patterns in Factorizations

As you practise, useful patterns start to jump out. Even numbers always begin with a factor of 2, and numbers ending in 0 or 5 always include a 5. Numbers whose digits add up to a multiple of 3 always include a 3, so 84, whose digits add to 12, is divisible by 3. Perfect squares such as 36, 100, and 144 have only even exponents in their factorization. Recognising these patterns lets you predict the first few primes before you even start dividing, which is exactly the kind of number sense that makes school and competitive maths feel effortless.

A Short Practice Set

To build fluency, try factorising these on your own and then check them: 50, 96, 125, and 150. Working through, 50 is 2 × 5², 96 is 2⁵ × 3, 125 is 5³, and 150 is 2 × 3 × 5². Notice how 125, a power of a single prime, has a very simple factorization, while 96, a highly even number, is dominated by twos. Doing a handful of these each day quickly turns prime factorization from a chore into a reflex, and it prepares you for the HCF, LCM, and fraction questions that rely on it.

Why These Numbers Matter

The numbers in this article are not random; they are the ones that recur throughout the Indian curriculum. Values like 24, 36, 60, 72, 120, and 360 appear constantly in geometry, time, and mensuration problems because they have many divisors and split neatly. Becoming instantly familiar with their prime factorizations means you can tackle a large share of textbook and exam questions without pausing to work from scratch, saving precious time when it matters most.

Putting It All Together

Prime factorization is a skill that rewards steady practice more than memorisation. Start with the smallest prime that fits, keep dividing until only primes remain, write the answer in exponent form, and always check by multiplying back. Do this with a handful of numbers each day, lean on divisibility rules to choose your primes, and use a calculator only to confirm your working. Before long, factorising numbers up to a few hundred will feel automatic, and the HCF, LCM, and fraction questions that depend on it will become some of the easiest marks in your exam.

Frequently Asked Questions

What is the prime factorization of 24?
24 = 2 × 2 × 2 × 3, written neatly as 2³ × 3. You reach it by dividing 24 by 2 three times to get 3, which is prime.

What is the prime factorization of 84?
84 = 2 × 2 × 3 × 7, or 2² × 3 × 7. Divide by 2 twice to get 21, then by 3 to get 7, which is prime.

How do I know which prime to start with?
Always start with the smallest prime that divides the number. If the number is even, start with 2; if it is odd, try 3, then 5, then 7, and so on.

Why do perfect squares have even exponents?
A perfect square is a number multiplied by itself, so every prime in its factorization appears an even number of times, giving even exponents like in 36 = 2² × 3².

How can I check my prime factorization?
Multiply all the prime factors back together. If the product equals your original number, your factorization is correct.