Divisibility and primes
Which whole numbers divide a given number exactly, and what is every whole number made of?
Where you are 10 inside
What this area is about
One whole number divides another when the division leaves remainder 0. Asking which numbers do that sorts the whole numbers greater than 1 into primes, with exactly two factors, and composites, with more. Every composite number then breaks into primes in exactly one way, apart from order.
How these fit together
Definitions. Finding all factors lists the numbers that divide a number, and listing multiples goes the other way. Prime and composite numbers sorts numbers by how many factors they have, and perfect, abundant, deficient and amicable numbers by the sum of their factors other than themselves.
Shortcuts. Divisibility rules read the last digit for 2, 5 and 10; tests for three and nine read the digit sum. The sieve of Eratosthenes finds all primes up to a limit at once.
Procedures. Prime factorization splits a number until only primes remain, and greatest common factor and least common multiple find what two numbers share.
The result. Every whole number greater than 1 has exactly one prime factorization, apart from order.
A path through it
Factors first. Then multiples, the divisibility rules (two, five and ten before three and nine), primes and composites, and the perfect and amicable numbers. Primes come before the sieve and prime factorization; factorization and multiples come before the greatest common factor and least common multiple, and factorization before uniqueness.
Next come common denominators, simplifying square roots and the Euclidean algorithm.