CBSE Class 10 Mathematics

## Primes and composites A **prime number** is a natural number greater than 1 whose only factors are 1 and itself. A **composite number** is a natural number greater than 1 that has at least one other factor, so it can be written as a product of smaller natural numbers. The number 1 is neither: it is left out of the primes so that factorisation is unique, because if 1 counted as a prime then $6 = 2 \times 3 = 1 \times 2 \times 3 = 1 \times 1 \times 2 \times 3$ would have infinitely many different factorisations. The number 2 is the only even prime, since every other even number has 2 as a factor. Two positive integers are **coprime** (or relatively prime) when their only common factor is 1. The numbers 8 and 15 are coprime even though neither is prime, and two different primes are always coprime. ## The fundamental theorem of arithmetic > Every composite number can be expressed as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur. "Unique apart from the order" means that $360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5$ can also be written $5 \times 3 \times 2 \times 3 \times 2 \times 2$, but no other collection of primes multiplies to 360. The usual way to write a factorisation is with the primes in increasing order and repeated primes as powers: $$ 360 = 2^3 \times 3^2 \times 5 $$ To find it, divide repeatedly by the smallest prime that goes in, starting with 2, and continue until the quotient is 1. For…