MP Board · Class 10 · Mathematics · Real NumbersExplain the Fundamental Theorem of Arithmetic in detail. Discuss its uniqueness and its various applications in the field of mathematics, specifically regarding the properties of numbers.

The Fundamental Theorem of Arithmetic

1. Definition and Statement:\nThe Fundamental Theorem of Arithmetic states that 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. In simpler terms, any integer greater than 1 is either a prime number itself or can be represented as a product of prime numbers.

2. The Concept of Uniqueness:\nThe term 'unique' signifies that for any given composite number, there is only one set of prime factors that, when multiplied, result in that number. For example, the number 30 can be written as $2 \times 3 \times 5$. While the order can change (e.g., $5 \times 2 \times 3$), the set of primes {2, 3, 5} remains constant. This uniqueness is vital because it provides a 'DNA' for every composite number.

3. Significance of the Order of Factors:\nWhile the theorem states the factorization is unique, it allows for the factors to be rearranged. Mathematically, we usually write the prime factorization in ascending order (e.g., $2^2 \times 3 \times 5^2$) to maintain a standard form, which helps in comparing different numbers.

4. Applications in Mathematics:

• Finding HCF and LCM: The theorem is the basis for finding the Highest Common Factor (HCF) and Least Common Multiple (LCM) of two or more numbers by comparing the powers of their prime factors.
• Proving Irrationality: It is used to prove that numbers like $\sqrt{2}$, $\sqrt{3}$, and $\sqrt{5}$ are irrational. The proof relies on the fact that if a prime $p$ divides $a^2$, then $p$ must also divide $a$.
• Decimal Expansions: It helps determine whether the decimal expansion of a rational number is terminating or non-terminating repeating. If the prime factors of the denominator are of the form $2^n 5^m$, the decimal terminates.
• Number Properties: It helps in checking if a number like $4^n$ or $6^n$ can ever end with the digit zero for any natural number $n$.