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what is greatest common factor

The greatest common factor (GCF) is the largest number that divides two or more whole numbers exactly, with no remainder.

Quick Scoop: What Is Greatest Common Factor?

Simple definition

  • The greatest common factor of two or more numbers is the biggest whole number that is a factor of all those numbers.
  • It is also called highest common factor (HCF) or greatest common divisor (GCD) in some books.

In plain words: list all the numbers that divide each number evenly; the largest one they share is the GCF.

Quick examples

  1. GCF of 12 and 16
    • Factors of 12: 1, 2, 3, 4, 6, 12
 * Factors of 16: 1, 2, 4, 8, 16

 * Common factors: 1, 2, 4

 * Greatest common factor: **4**
  1. GCF of 18 and 21
    • 18 = 2×3×32\times 3\times 32×3×3
 * 21 = 3×73\times 73×7

 * Common prime factor: 3

 * Greatest common factor: **3**
  1. GCF of 20, 50, and 120
    • Common factors: 1, 2, 5, 10
 * Greatest common factor: **10**

Why GCF matters (real use)

  • Reducing fractions : To simplify 1824\frac{18}{24}2418​, you divide top and bottom by the GCF, which is 6, giving 34\frac{3}{4}43​.
  • Sharing or grouping : If you want to split objects into equal groups with no leftovers, the GCF tells you the largest possible group size.
  • Algebra & factoring: In algebra, you often “factor out” the GCF from terms to simplify expressions.

Three common ways to find the GCF

  1. List of factors (good for small numbers)
 * List all factors of each number.
 * Circle the common ones.
 * Pick the largest.
  1. Prime factorization (more systematic)
 * Break each number into prime factors.
 * Take only the primes they all share, with the smallest exponents.
 * Multiply those shared primes to get the GCF.
  1. Euclidean algorithm (best for big numbers)
    • Repeatedly divide and take remainders until the remainder is 0; the last nonzero remainder is the GCF.

Mini table: concept at a glance (HTML as requested)

[1] [3] [5]
Numbers Method Work GCF
12, 16 List of factors Common factors: 1, 2, 44
18, 21 Prime factorization 18 = 2×3×3, 21 = 3×7, common: 33
20, 50, 120 List of factors / primes Common factors: 1, 2, 5, 1010

Tiny story to remember it

Imagine three friends bringing snacks to share in class: one has 20 cookies, one has 50, one has 120. They want to make identical snack bags with no crumbs left over. They ask, “What is the greatest number of bags we can make so that we split everything evenly?” The answer to that question is exactly the greatest common factor of 20, 50, and 120.

TL;DR: The greatest common factor is the biggest number that divides all your numbers evenly, used to simplify fractions and share things fairly.

Information gathered from public forums or data available on the internet and portrayed here.