Mathematics.

number theory basics

Factors and Multiples

Pre-Algebra25 minDifficulty2 out of 10

You should know: integers

Overview

A factor of a whole number n is a whole number that divides n exactly, with no remainder. A multiple of n is any number obtained by multiplying n by a whole number. These two ideas are mirror images of each other: 4 is a factor of 12 because 12 ÷ 4 has no remainder, and 12 is a multiple of 4 because 12 = 4 × 3. Factors and multiples underlie fraction simplification, finding common denominators, and scheduling problems (like figuring out when two repeating events coincide).

Intuition

If you arrange 12 objects into equal rows, the possible row lengths (1, 2, 3, 4, 6, 12) are exactly the factors of 12. Multiples run the other direction: starting from 4, the multiples are 4, 8, 12, 16, 20, ... — every number you land on if you keep counting by 4s. The greatest common factor (GCF) of two numbers is the largest number that divides both; the least common multiple (LCM) is the smallest number that both divide into.

Formal Definition

Definition

For integers a and n, a is a factor (or divisor) of n if there exists an integer k such that n = ak. Equivalently, n is a multiple of a. The greatest common factor (GCF) of a and b is the largest positive integer dividing both; the least common multiple (LCM) is the smallest positive integer that both a and b divide.

n=akfor some integer kn = ak \quad \text{for some integer } k
Divisibility
gcd⁡(a,b)×lcm⁡(a,b)=a×b\gcd(a,b) \times \operatorname{lcm}(a,b) = a \times b

Holds for positive integers a, b

GCF-LCM relation

Properties

Every number is a factor of itself

n∣nn \mid n

1 divides everything

1∣n for all integers n1 \mid n \text{ for all integers } n

GCF-LCM product rule

gcd⁡(a,b)⋅lcm⁡(a,b)=ab\gcd(a,b)\cdot\operatorname{lcm}(a,b) = ab

Applications

GCF computations (via the Euclidean algorithm) underlie fraction simplification and cryptographic algorithms like RSA key generation.

Worked Examples

  1. Find every whole number that divides 18 with no remainder.

    18=1×18=2×9=3×618 = 1\times18 = 2\times9 = 3\times6

Answer: 1, 2, 3, 6, 9, 18

Practice Problems

Difficulty 1/10

List the first five multiples of 7.

Difficulty 3/10

Find the GCF of 24 and 36.

Common Mistakes

Common Mistake

Confusing factors and multiples — e.g. saying '4 is a multiple of 12' when 4 is actually a factor of 12.

Factors of n are smaller than or equal to n and divide it; multiples of n are n scaled up (n, 2n, 3n, ...) and are greater than or equal to n.

Common Mistake

Assuming the GCF of two numbers is always small or the LCM is always their product.

The LCM equals the product only when the two numbers share no common factors besides 1 (i.e. gcd = 1). In general, lcm(a,b) = ab / gcd(a,b).

Summary

  • A factor of n divides n exactly; a multiple of n is n times a whole number.
  • The greatest common factor (GCF) is the largest shared factor of two numbers.
  • The least common multiple (LCM) is the smallest number both are factors of.
  • GCF and LCM are related by gcd(a,b) × lcm(a,b) = ab.
  • Factors and multiples are foundational to simplifying fractions and finding common denominators.

References

  1. WebsiteWikipedia — Divisor