Fundamental Theorem of Arithmetic from "summary" of NCERT Class 10 Mathematics Solutions by JagranJosh
The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of prime numbers, and this factorization is unique up to the order of the factors. In other words, any composite number can be written as a unique product of prime numbers. This theorem is fundamental in number theory and forms the basis for many mathematical concepts. It helps us understand the structure of numbers and their relationships with each other. The prime factorization of a number is crucial in various mathematical calculations, including finding the greatest common divisor or least common multiple of two numbers. For example, let's consider the number 24. It can be expressed as 2 * 2 * 2 * 3, where 2 and 3 are prime numbers. This prime factorization is unique, and any other factorization of 24 will involve these same prime numbers. The Fundamental Theorem of Arithmetic ensures that we can uniquely decompose any composite number into its prime factors, providing a way to understand the properties of numbers and their interactions. This theorem is essential in various mathematical fields, such as cryptography, number theory, and algebra. Understanding the Fundamental Theorem of Arithmetic helps us appreciate the beauty and structure of numbers. It allows us to solve complex problems involving factorization and helps us explore the properties of numbers in a systematic way. This theorem is a cornerstone of mathematics and plays a vital role in shaping our understanding of numbers and their properties.
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