Divisibility and Factors

We say that one whole number a divides another whole number b, written a divides b, if b can be expressed as a times some other whole number with no remainder. For example, 4 divides 12 because 12 equals 4 times 3. Five does not divide 12 because 12 divided by 5 leaves a remainder of 2. The numbers that divide a given integer are called its divisors or factors. The factors of 12 are 1, 2, 3, 4, 6, and 12.

Quick divisibility rules let you check by inspection. A number is divisible by 2 if its last digit is even. By 3 if the sum of its digits is divisible by 3. By 4 if the last two digits form a number divisible by 4. By 5 if it ends in 0 or 5. By 9 if the sum of its digits is divisible by 9. By 10 if it ends in 0. These rules look like tricks, but each follows from how the base-10 number system encodes powers of ten.

Two numbers are called common factors of a third number if they both divide it. Two numbers share common factors when they have any divisor in common other than 1. The greatest common factor of 12 and 18 is 6. Two numbers whose greatest common factor is 1 are called relatively prime or coprime. For example, 8 and 15 are coprime, even though neither one is itself prime. Coprimality is an essential building block for ratios, modular arithmetic, and cryptography.

Divisibility seems elementary, but it is the workhorse of number theory. Concepts that look advanced, like prime factorization, modular arithmetic, and even cryptographic keys, all reduce to questions about who divides what. Once you can spot factor structure quickly, much of higher arithmetic becomes pattern recognition.