Fundamental Theorem of Arithmetic Class 10 – Easy Explanation, Notes & Examples

 The Fundamental Theorem of Arithmetic – CBSE Class 10 

πŸ” What You Already Know

From earlier classes, you know that:

  • Every natural number can be written as a product of prime numbers.
  • Example:
    • 2 = 2
    • 4 = 2 × 2
    • 253 = 11 × 23

But now, we ask the reverse:

Can every natural number be made by multiplying prime numbers?

πŸ“Œ Understanding with Examples

Take a set of primes: 2, 3, 7, 11, 23
Now multiply some or all of them:

  • 7 × 11 × 23 = 1771
  • 2 × 3 × 7 × 11 × 23 = 10626
  • 3 × 7 × 11 × 23 = 5313
  • 2² × 3 × 7 × 11 × 23 = 21252

You can create infinitely many such numbers using combinations and powers of primes.

πŸ“˜ Important Idea

  • There are infinitely many prime numbers.
  • All composite numbers are made by multiplying primes.

But here's the big question:

Is it possible that some composite number cannot be expressed as a product of primes?

Let’s explore by factorising numbers.

🌳 Factor Tree Method

Let’s take a number and break it into its prime factors using a factor tree.

Example: 32760

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

Another Example: 123456789
123456789 = 3² × 3803 × 3607
(Both 3803 and 3607 are prime numbers – you can check!)

✅ Conjecture (Logical Guess)

Every composite number can be written as a product of powers of primes.

This statement has been proved to be true and is one of the most important theorems in mathematics.

⭐ The Fundamental Theorem of Arithmetic – Statement

Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, except for the order of the prime factors.

🧾 Key Points to Remember

  • Every natural number > 1 is either:
    • A prime itself, or
    • A product of prime numbers
  • The prime factorisation is unique, meaning:
    • 60 = 2² × 3 × 5
    • No other set of primes will give 60 in a different way (only the order can change)
  • Order doesn’t matter:
    2 × 3 × 5 × 2 = 2² × 3 × 5 = 60

🧠 Why is it Important?

  • Forms the basis of arithmetic and number theory
  • Used in:
    • Finding HCF and LCM
    • Solving Diophantine equations
    • Cryptography and computer algorithms

πŸ’‘ Quick Revision

Concept Meaning
Prime Numbers Numbers greater than 1 that have no divisors other than 1 and itself
Composite Numbers Numbers that are not prime; they can be written as a product of primes
Unique Factorisation Every number has only one set of prime factors (excluding order)

πŸ“ Practice Questions

1. Very Short Answer

Q. Write the prime factorisation of 100.
A. 100 = 2² × 5²

2. Short Answer

Q. Is 2 × 2 × 3 × 5 a unique prime factorisation of 60?
A. Yes, because 2² × 3 × 5 = 60 and the order does not affect uniqueness.

3. Application

Q. Write the Fundamental Theorem of Arithmetic and explain with an example.
A.
Statement: Every composite number can be expressed as a product of prime numbers in a unique way (order may vary).
Example: 90 = 2 × 3² × 5

🧠 Summary

  • Every natural number > 1 is either a prime or a product of primes.
  • This factorisation is unique, and forms the core of arithmetic.
  • This is called the Fundamental Theorem of Arithmetic.


Comments

Post a Comment

Popular posts from this blog

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

Class 10 Maths – Theorem : Fundamental Theorem of Arithmetic 🧠 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. " ✅ 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. πŸ”’ 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 πŸ‘‰ It doesn’t matter — it’s still the same set of prime numbers multiplied together. ✅ That’s what “ unique, except for order ” means. πŸ’‘ Important Notes A prime factorisation is a way to express a ...
Read more

Class 10 Maths Chapter 1 Real Numbers Introduction – Euclid’s Division Algorithm & Fundamental Theorem Explained

Real Numbers – Introduction CBSE Class 10 Maths πŸ” What You Already Know (from Class 9) : Real Numbers = Rational + Irrational numbers Rational Numbers : Can be written as p/q (q ≠ 0) Irrational Numbers : Cannot be written as p/q Examples: √2, √5, Ο€ 🧠 What You Will Learn in This Chapter : In Class 10, we go deeper into the properties of real numbers, especially positive integers , through: Euclid’s Division Algorithm Fundamental Theorem of Arithmetic These concepts form the foundation of number theory , with applications in HCF, irrational numbers, and decimal expansions. πŸ“1. Euclid’s Division Algorithm πŸ“Œ Definition : Given two positive integers a and b (a > b), there exist integers q and r such that: a = bq + r {where } 0≤r< b This is basically the long division method you already know! πŸ”§ Key Use : Helps us find the HCF (Highest Common Factor) of two numbers. Example: To find HCF of 210 and 45, we apply Euclid’s algorithm step by s...
Read more