Theorem: Fundamental Theorem of Arithmetic Explained with Examples – CBSE Class 10 Maths Notes

Class 10 Maths – Theorem : Fundamental Theorem of Arithmetic

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🧠 What Does the Theorem Say?

"Every composite number can be expressed (factorised) as a product of prime numbers, and this factorisation is unique, apart from the order of the prime factors."

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✅ Let’s Break It Down:

• A composite number is a natural number that is not prime and has more than two factors.
• According to the theorem:
  • Every composite number can be written as a product of prime numbers.
  • This factorisation is unique, except for the order in which the primes are written.

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🔢 Example

Take the number 32760

We can write:

32760 = 2 × 2 × 2 × 3 × 3 × 5 × 7 × 13
= 2³ × 3² × 5 × 7 × 13

Now, whether you write it as:
2 × 3 × 5 × 7 × 13 × 2 × 3 × 2
or
3 × 2 × 2 × 5 × 7 × 3 × 2 × 13
👉 It doesn’t matter — it’s still the same set of prime numbers multiplied together.

✅ That’s what “unique, except for order” means.

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💡 Important Notes

• A prime factorisation is a way to express a number using only prime numbers multiplied together.
• Every number has only one such prime factorisation.
• Order doesn’t matter – 2 × 3 × 5 = 5 × 3 × 2
• To avoid confusion, we usually write prime factors in ascending order.

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📘 In General:

For any composite number x:

x = p₁ × p₂ × ... × pₙ
Where each p is a prime number, and p₁ ≤ p₂ ≤ ... ≤ pₙ

If any prime number repeats, it is written as a power:

E.g. 2 × 2 × 2 = 2³

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📌 Why Is It Called ‘Fundamental’?

Because this theorem is the foundation of:

• Prime factorisation
• Finding HCF and LCM
• Cryptography and computer science
• Advanced number theory

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📚 Applications of the Theorem

✅ 1. Finding HCF and LCM

We use prime factorisation to find the Highest Common Factor (HCF) and Least Common Multiple (LCM) of two or more numbers.

✅ 2. Solving Number Puzzles

Many math problems use the idea that numbers are built from unique prime blocks.

✅ 3. Cryptography (Cyber Security)

Modern computer encryption depends on the fact that large numbers are hard to factor into primes.

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📝 Practice Example

Q. Factorise 90 using the Fundamental Theorem of Arithmetic.

Solution:
90 = 2 × 3 × 3 × 5 = 2 × 3² × 5
👉 Unique prime factorisation (apart from order)

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✨ Quick Recap

Term	Meaning
Composite Number	A number with more than 2 factors
Prime Number	A number with only 2 factors: 1 and itself
Prime Factorisation	Breaking a number into product of prime numbers
Theorem Statement	Every composite number has a unique prime factorisation, ignoring order

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🔁 Remember

Once we fix the order (like ascending), the factorisation becomes completely unique.

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