Fundamental Theorem of Arithmetic Hindi NCERTCBSE 10th Class Maths

Understanding the Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic is one of the most important results in number theory. It states that every composite number can be expressed as a product of prime numbers, and this factorisation is unique, apart from the order of the prime factors. In simpler terms, any composite number you pick can be broken down into prime factors in exactly one way. For example, the number 12 can be written as 2 × 2 × 3, and there is no other combination of prime numbers that will multiply to give you 12. This property holds true for all composite numbers, no matter how large or small they are.

To understand this theorem better, let us recall what prime and composite numbers are. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Examples include 2, 3, 5, 7, 11, and so on. A composite number, on the other hand, is a natural number greater than 1 that has at least one divisor other than 1 and itself. Examples include 4, 6, 8, 9, 10, and so forth. When we factorise a composite number completely, we keep breaking it down until all the factors are prime. This process is called prime factorisation. For instance, take the number 156. We can factorise it as 156 = 2 × 78 = 2 × 2 × 39 = 2 × 2 × 3 × 13. Here, 2, 3, and 13 are all prime numbers, so the prime factorisation of 156 is 2² × 3 × 13. No matter what method you use to factorise 156, you will always arrive at this same set of prime factors.

One of the most useful applications of the Fundamental Theorem of Arithmetic is in finding the Highest Common Factor (HCF) and Least Common Multiple (LCM) of two or more numbers. By expressing each number as a product of prime powers, we can easily determine the HCF by taking the lowest power of each common prime factor, and the LCM by taking the highest power of each prime factor that appears. This method is systematic and avoids the confusion that sometimes arises from other methods. Additionally, the theorem helps us understand why 1 is neither prime nor composite — if 1 were considered prime, the uniqueness of prime factorisation would break down, since we could insert any number of 1s into a factorisation without changing the product.

• Every composite number has a unique prime factorisation, apart from the order of factors.
• Prime factorisation is the process of expressing a number as a product of its prime factors.
• HCF of two or more numbers can be found by taking the product of the lowest powers of all common prime factors.
• LCM of two or more numbers can be found by taking the product of the highest powers of all prime factors involved.
• The relationship HCF(a, b) × LCM(a, b) = a × b holds true for any two positive integers.
• The number 1 is neither prime nor composite, which is essential for maintaining the uniqueness of the theorem.

The Fundamental Theorem of Arithmetic forms a core part of the CBSE Class 10 Mathematics chapter on Real Numbers. It connects directly to the concepts of HCF and LCM, which students have studied in earlier classes, but now provides a more rigorous and logical foundation. Understanding this theorem thoroughly will help students solve problems related to divisibility, prime factorisation, and number properties — all of which are frequently tested in board examinations and are essential building blocks for higher-level mathematics.