Factors, multiples and primes

Builds on

Required prerequisites, all the way down: 4

You will learn to

• List the factors of numbers up to 100 and find common factors and common multiples.
• Classify numbers as prime or composite and use the sieve of Eratosthenes.
• Write a whole number as a product of primes using a factor tree, and explain the divisibility tests for 2, 3, 5, 9 and 10.

Definition

• b is a factor of a, and a is a multiple of b, if a = b × k for some whole number k.
• A prime is a whole number greater than 1 whose only factors are 1 and itself; a composite number is a whole number greater than 1 that is not prime.

Theorem

• Every whole number greater than 1 is a prime or a product of primes, and the product is unique apart from order (the fundamental theorem of arithmetic, proved later).
• A number is divisible by 3 (or by 9) exactly when the sum of its digits is.
• 1 is neither prime nor composite.

Common misconceptions

• Thinking 1 is prime.
• Believing all odd numbers are prime, although 9 = 3 × 3.
• Confusing factors with multiples.

Leads to

Optional discoveries

• Cryptography RSA encryption relies on the belief, widely held but not proved, that factoring very large numbers is computationally hard.
• Rings, ideals and fields Unique factorisation can fail in other number systems: in ℤ[√−5], 6 = 2 · 3 = (1 + √−5)(1 − √−5).
• Divisibility, Euclid's algorithm and unique factorisation 2, 3, 5, 7, 11, 13, … The primes get rarer as the numbers grow. Do they ever run out? And how rare do they get?Wonder Trail →

Further reading

• Hans Magnus Enzensberger, The Number Devil