Greatest Common Factor Formula

The Formula

\text{GCF}(a, b) \times \text{LCM}(a, b) = a \times b

When to use: The biggest 'piece' size that fits evenly into two numbers—like the largest tile that covers both a 12-unit and 18-unit floor.

Quick Example

GCF of 12 and 18: Factors of 12 (1,2,3,4,6,12) and 18 (1,2,3,6,9,18). GCF = 6.

Notation

\text{GCF}(a, b) or \gcd(a, b) denotes the greatest common factor of a and b

What This Formula Means

The largest positive integer that divides evenly into two or more given numbers with no remainder.

The biggest 'piece' size that fits evenly into two numbers—like the largest tile that covers both a 12-unit and 18-unit floor.

Formal View

\gcd(a, b) = \max\{d \in \mathbb{Z}^+ : d \mid a \text{ and } d \mid b\}. Via prime factorization: if a = \prod p_i^{\alpha_i} and b = \prod p_i^{\beta_i}, then \gcd(a,b) = \prod p_i^{\min(\alpha_i, \beta_i)}.

Worked Examples

Example 1

easy
Find the GCF of 48 and 36.

Solution

  1. 1
    Prime-factor each number: 48 = 2^4 \times 3 and 36 = 2^2 \times 3^2.
  2. 2
    Identify the common prime factors and keep the smaller exponent for each: 2^2 and 3^1.
  3. 3
    Multiply those shared factors: 2^2 \times 3 = 4 \times 3 = 12, so the GCF is 12.

Answer

12
The GCF is the product of shared prime factors, each raised to the lowest power appearing in either factorization. The GCF is useful for simplifying fractions.

Example 2

medium
Find the GCF of 84, 126, and 210.

Common Mistakes

  • Confusing GCF with LCM — GCF of 12 and 18 is 6 (largest common factor), while LCM is 36 (smallest common multiple)
  • Taking the larger power of each prime instead of the smaller — for 12 = 2^2 \times 3 and 18 = 2 \times 3^2, the GCF uses 2^1 and 3^1, giving 6, not 2^2 \times 3^2 = 36
  • Stopping at the first common factor found — finding that 2 divides both 12 and 18, but not checking for the greatest: 6 is the GCF, not 2

Why This Formula Matters

Essential for reducing fractions to lowest terms: \frac{12}{18} = \frac{2}{3} because \gcd(12,18) = 6.

Frequently Asked Questions

What is the Greatest Common Factor formula?

The largest positive integer that divides evenly into two or more given numbers with no remainder.

How do you use the Greatest Common Factor formula?

The biggest 'piece' size that fits evenly into two numbers—like the largest tile that covers both a 12-unit and 18-unit floor.

What do the symbols mean in the Greatest Common Factor formula?

\text{GCF}(a, b) or \gcd(a, b) denotes the greatest common factor of a and b

Why is the Greatest Common Factor formula important in Math?

Essential for reducing fractions to lowest terms: \frac{12}{18} = \frac{2}{3} because \gcd(12,18) = 6.

What do students get wrong about Greatest Common Factor?

Using prime factorization: GCF uses the smaller power of each common prime.

What should I learn before the Greatest Common Factor formula?

Before studying the Greatest Common Factor formula, you should understand: factors, divisibility intuition.

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