NCERT · Class 10 · Mathematics · Real NumbersExplain the Fundamental Theorem of Arithmetic in detail. Discuss its significance in number theory and describe at least three of its major applications in mathematics. (5 Marks)

The Fundamental Theorem of Arithmetic

\nThe Fundamental Theorem of Arithmetic is a foundational principle in number theory. It states: "Every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur."

1. Understanding the Theorem\nA composite number is a positive integer greater than 1 that has at least one divisor other than 1 and itself. The theorem asserts that any such number can be broken down into prime building blocks. For example, the number 120 can be written as $2 \times 2 \times 2 \times 3 \times 5$ or $2^3 \times 3 \times 5$. No matter how you perform the factorization, you will always get three 2s, one 3, and one 5.

2. Significance in Mathematics

• Uniqueness: The uniqueness property ensures that every integer has a unique 'prime signature'. This allows mathematicians to study the properties of numbers by analyzing their prime factors.
• Structural Basis: It forms the basis for understanding the divisibility properties of integers, which is essential for higher-level algebra and number theory.

3. Major Applications

• Finding HCF and LCM: The theorem provides a systematic method to calculate the Highest Common Factor (HCF) and Least Common Multiple (LCM). HCF is the product of the smallest power of each common prime factor, while LCM is the product of the greatest power of every prime factor involved.
• Proving Irrationality: This theorem is used to prove that numbers like $\sqrt{2}, \sqrt{3},$ and $\sqrt{5}$ are irrational. By assuming they are rational ($p/q$), we use prime factorization to show a contradiction regarding the common factors of $p$ and $q$.
• Decimal Expansions: It helps determine if a rational number $p/q$ has a terminating or non-terminating repeating decimal expansion. If the prime factorization of the denominator $q$ is of the form $2^n 5^m$, the decimal terminates.

Conclusion\nIn conclusion, the Fundamental Theorem of Arithmetic is vital because it links the additive nature of integers with their multiplicative structure, serving as a bridge to advanced mathematical concepts like cryptography and complex analysis.