Worked Example • Half-Dozen & Dozen

LCM of 6 and 12 (Equals 12)

The Least Common Multiple of 6 and 12 is 12. Since 6 divides evenly into 12 (12 = 6 × 2), 12 is already a multiple of 6, making 12 the LCM and 6 the GCF.

LCM of 6 and 12 Result

LCM(6, 12) = 12

Greatest Common Factor: GCF(6, 12) = 6

Check the answer (every division leaves a remainder of 0):

12 ÷ 6 = 212 ÷ 12 = 1

Show the Working (Step-by-Step)

Switch method in one click:

Step 1: Break each number down into its prime factors.

Write each input as a product of prime numbers, then group matching primes with exponents.

6 = 2 × 3

= 2 × 3

12 = 2 × 2 × 3

= 2² × 3

Step 2: For each prime, pick the highest exponent that appears in any single number.

Prime	In 6	In 12	Highest Power
2	2¹	2²	2² = 4
3	3¹	3¹	3¹ = 3

Step 3: Multiply those highest prime powers together to get the LCM.

LCM = 2² × 3¹ = 4 × 3 = 12

Full Written Solution for the LCM of 6 and 12

If you are copying the steps onto a homework worksheet, here is how each method reaches LCM(6, 12) = 12:

Method A: Prime Factorization

• 6 = 2 × 3
• 12 = 2² × 3

Multiply the highest power of each prime: 2² × 3¹ = 12.

Method B: GCF & Exact Division Check

The Greatest Common Factor of 6 and 12 is 6.

12 ÷ 6 = 2 (remainder 0 ✓)

12 ÷ 12 = 1 (remainder 0 ✓)

Real-World & Fraction Context

Egg cartons of 6 (half-dozen) and 12 (full dozen) match up at 12 eggs—two small cartons equal one standard carton.

Mistake to Watch Out For

Never write 6 × 12 = 72, and don't mix up GCF(6, 12) = 6 with LCM(6, 12) = 12.