Introduction to Factors and Multiples

Every number has hidden structure. Understanding factors and multiples unlocks that structure and makes many areas of mathematics — from simplifying fractions to solving equations — much easier to handle.

These ideas are genuinely ancient. The Greek mathematician Euclid formalised the concept of one number “measuring” another — precisely what we now call a factor — in Book VII of his Elements, written around 300 BCE, the same text that also contains the earliest known algorithm for finding the greatest common factor. A few decades later, around 240 BCE, the Alexandrian scholar Eratosthenes devised his famous “sieve” for filtering prime numbers out of a list of consecutive integers — a method still taught almost unchanged more than 2,200 years later. Two thousand years after that, in 1801, the German mathematician Carl Friedrich Gauss proved in his Disquisitiones Arithmeticae that every whole number greater than 1 breaks down into prime factors in exactly one way — the Fundamental Theorem of Arithmetic that quietly underpins everything in this learning path.

What This Learning Path Covers

A Quick Preview

The number 12 can be divided exactly by 1, 2, 3, 4, 6, and 12. These are its factors. The numbers 12, 24, 36, 48... are its multiples — numbers you get by multiplying 12 by 1, 2, 3, 4, and so on. The number 12 itself can be broken into prime pieces: 2 × 2 × 3. This is its prime factorization.

Why Does It Matter?

• Simplifying fractions: 12/18 = 2/3 (dividing by GCF 6)
• Adding fractions: 1/4 + 1/6 needs LCM 12 as common denominator
• Solving word problems about equal sharing, repeating events, and scheduling
• Cryptography and computer science rely heavily on prime factorization

Learning Objectives

Where to Start

Click Next below to begin with factors — the foundation of everything in this section.

Quick Check: Factor or Multiple?