Find the greatest common factor of 12, 18 and 24.

How we read your problem

Find the greatest common factor (also called GCD or HCF) of the numbers.

Step-by-step solution

1. 1

   Prime factorization of each number

   $$12 = 2^{2} \times 3,\quad 18 = 2 \times 3^{2},\quad 24 = 2^{3} \times 3$$
2. 2

   Euclidean algorithm for GCF(12, 18)

   18 = 1 × 12 + 6; 12 = 2 × 6 + 0

   $$18 = 1 \times 12 + 6 \\ 12 = 2 \times 6 + 0$$
3. 3

   The last non-zero remainder is the GCF

   $$\gcd(12, 18) = 6$$
4. 4

   Euclidean algorithm for GCF(6, 24)

   24 = 4 × 6 + 0

   $$24 = 4 \times 6 + 0$$
5. 5

   The last non-zero remainder is the GCF

   $$\gcd(6, 24) = 6$$
6. 6

   Confirm with shared prime factors

   Multiply the primes common to every number, each with its smallest exponent.

   $$2 \times 3 = 6$$

Formulas used

• Euclidean algorithm

  $$\gcd(a, b) = \gcd(b,\ a \bmod b),\quad \gcd(a, 0) = a$$

6 is the largest whole number that divides every number exactly.

Verification

• 12 ÷ 6 = 2
• 18 ÷ 6 = 3
• 24 ÷ 6 = 4