Real Numbers | Class 10 Mathematics Notes

1.2 The Fundamental Theorem of Arithmetic

This section elaborates on the Fundamental Theorem of Arithmetic, which states that every composite number can be expressed as a product of prime numbers in a unique way, except for the order of the factors. The section begins by exploring the idea of constructing numbers by multiplying prime numbers, illustrating with examples such as 7 × 11 × 23 = 1771 and 2³ × 3 × 7³ = 8232. It then discusses the infinite nature of prime numbers and the question of whether every composite number can be expressed as a product of primes. Using factor trees, the section demonstrates the factorization of numbers like 32760 and 123456789 into their prime factors, confirming the conjecture that every composite number can indeed be factorized uniquely into primes. The theorem is formally stated and its historical context is given, highlighting Carl Friedrich Gauss's contribution. The uniqueness of prime factorization is emphasized, meaning that while the order of factors can vary, the prime factors themselves and their powers are unique. The section also revisits the method of finding HCF and LCM using prime factorization, illustrating with examples such as finding the HCF and LCM of 6 and 20, and 96 and 404. It concludes by noting that while the product of HCF and LCM equals the product of two numbers, this property does not extend straightforwardly to three or more numbers.

📊 Diagram: Figure showing factor tree for 32760; Figure of Carl Friedrich Gauss (1777 – 1855).

🧪 Activity: Exercises include prime factorization of numbers, finding HCF and LCM, and verifying properties of these.

🔗 Connection: Prepares for Section 1.3, where irrational numbers and their properties are revisited and proved using the Fundamental Theorem of Arithmetic.