Worked Example • Pairwise Overlap Triple

LCM of 12, 15, and 20 (Equals 60)

The Least Common Multiple of 12, 15, and 20 is 60. By prime factorization, 12 = 2² × 3, 15 = 3 × 5, and 20 = 2² × 5, so multiplying the highest prime powers gives 2² × 3¹ × 5¹ = 60.

LCM of 12, 15, and 20 Result

LCM(12, 15, 20) = 60

Greatest Common Factor: GCF(12, 15, 20) = 1No common factor greater than 1

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

60 ÷ 12 = 560 ÷ 15 = 460 ÷ 20 = 3

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.

12 = 2 × 2 × 3

= 2² × 3

15 = 3 × 5

= 3 × 5

20 = 2 × 2 × 5

= 2² × 5

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

Prime	In 12	In 15	In 20	Highest Power
2	2²	—	2²	2² = 4
3	3¹	3¹	—	3¹ = 3
5	—	5¹	5¹	5¹ = 5

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

LCM = 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60

Full Written Solution for the LCM of 12, 15, and 20

If you are copying the steps onto a homework worksheet, here is how each method reaches LCM(12, 15, 20) = 60:

Method A: Prime Factorization

• 12 = 2² × 3
• 15 = 3 × 5
• 20 = 2² × 5

Multiply the highest power of each prime: 2² × 3¹ × 5¹ = 60.

Method B: GCF & Exact Division Check

The Greatest Common Factor of 12, 15, and 20 is 1.

60 ÷ 12 = 5 (remainder 0 ✓)

60 ÷ 15 = 4 (remainder 0 ✓)

60 ÷ 20 = 3 (remainder 0 ✓)

Real-World & Fraction Context

Adding 1/12 + 1/15 + 1/20 (a classic optics and parallel-resistor equation: 1/f = 5/60 + 4/60 + 3/60 = 12/60 = 1/5) relies on LCD(12, 15, 20) = 60.

Mistake to Watch Out For

Seeing GCF = 1 and multiplying 12 × 15 × 20 = 3,600 is a classic algebra trap. Pairwise overlaps still reduce the LCM by a factor of 60!