Ch 5 of Ganita Prakash — "Prime Time" — introduces students to one of the most fundamental concepts in number theory: prime numbers. Students learn to distinguish between prime and composite numbers, find factors and multiples, apply divisibility rules, and discover the Sieve of Eratosthenes — a 2,000-year-old method for finding prime numbers.
A factor of a number divides it exactly without leaving a remainder. For example, factors of 12 are 1, 2, 3, 4, 6, 12. A multiple of a number is obtained by multiplying it by a whole number. Multiples of 3 are 3, 6, 9, 12, 15… Every number is a factor and multiple of itself.
A prime number has exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13. A composite number has more than two factors. Examples: 4, 6, 8, 9, 10, 12. The number 1 is special — it is neither prime nor composite because it has only one factor. The number 2 is the smallest and only even prime number.
The Sieve of Eratosthenes is an ancient algorithm to find all prime numbers up to a given limit. Write numbers from 2 to n. Start with 2 and cross out all its multiples. Move to the next un-crossed number (3) and cross out its multiples. Continue until you have processed all numbers up to √n. The remaining un-crossed numbers are prime.
Every composite number can be expressed as a product of prime numbers. This is called prime factorisation. For example: 36 = 2 × 2 × 3 × 3 = 2² × 3². To find the prime factorisation, keep dividing by the smallest prime factor until you reach 1. Prime factorisation is used to find HCF and LCM.
Quick rules help check divisibility: divisible by 2 if last digit is even; by 3 if digit sum is divisible by 3; by 5 if last digit is 0 or 5; by 9 if digit sum is divisible by 9; by 10 if last digit is 0. These rules help in quickly identifying factors of large numbers.
Download the official NCERT PDF for Ch 5 "Prime Time" from the NCERT website: https://ncert.nic.in/textbook/pdf/fegp105.pdf. You can also download the complete Ganita Prakash textbook: https://ncert.nic.in/textbook/pdf/fegp1ps.zip
A prime number must have exactly two distinct factors: 1 and itself. The number 1 has only one factor (1 itself), so it does not meet the definition of a prime number. This convention is important for the Fundamental Theorem of Arithmetic to work correctly.
Every even number greater than 2 is divisible by 2, which means it has at least three factors: 1, 2, and itself. Since prime numbers can only have two factors, no even number other than 2 can be prime.
It is an ancient algorithm (over 2,200 years old) devised by the Greek mathematician Eratosthenes to find all prime numbers up to a specified limit. You systematically eliminate multiples of each prime, starting with 2. Numbers that remain un-eliminated are prime.
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