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Prime Number

A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and the number itself. In the Class 10 Real Numbers curriculum, prime numbers are fundamental because every composite number can be uniquely expressed as a product of primes.

Key Concepts

The number 2 is the smallest prime number and the only even prime number; all other primes are odd.
According to the Fundamental Theorem of Arithmetic, the prime factorization of any composite number is unique, ignoring the order of the factors.
Prime factorization is the primary method used to calculate the Highest Common Factor (HCF) and Least Common Multiple (LCM) of positive integers.

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Common Misconceptions

Myth: The number 1 is considered a prime number.
Fact: The number 1 is neither prime nor composite because it has only one positive divisor, whereas a prime number must have exactly two distinct divisors.

Real World Applications

Modern computer cryptography, such as RSA encryption used in secure web browsing, relies on the difficulty of factoring the product of two large prime numbers.
Certain species of cicadas have evolutionary life cycles of 13 or 17 years, which are prime numbers, to minimize overlapping with the life cycles of their predators.

Frequently Asked Questions

What is the difference between prime and composite numbers?
A prime number has exactly two factors, 1 and itself, while a composite number has more than two factors.
How is prime factorization used to find HCF and LCM?
The HCF is the product of the smallest powers of each common prime factor, while the LCM is the product of the greatest powers of all prime factors involved in the numbers.

Keep Learning

Composite NumberPrime FactorizationFundamental Theorem of Arithmetic