Prime numbers are those that have 1 (one) and itself as the only two divisors. Examples of primes are 2, 3, 5, 7, 11. None of 4, 6, 9 is a prime since 4 = 2 × 2, 6 = 2 × 3, and 9 = 3 × 3.
If a number greater than one is not a prime, then it is a composite number, and can be factored into the product of primes — called prime factorization. We have given the prime factorization of 4, 6, 9 as above. For a couple of more examples:
12 = 2 × 2 × 3
36 = 2 × 3 × 3 × 3
28 = 2 × 2 × 7
So all natural numbers are divided into three classes: the number 1, the prime numbers, and the composite numbers.
A bonus point: π, besides representing in a circle, the ratio of circumference to diameter, also stands for a special function related to prime numbers. This is described as follows.
Function π(x) — for integer x, represents the number of primes less than or equal (i.e. not exceeding) x.
For example,
π(2) = 1, π(3) = 2, π(10) = 4, π(20) = 8 etc.
[To find why π(10) = 4, recall the 4 prime numbers not exceeding 10: they are 2,3,5, and 7.]
Problem 3