How To Find Least Common Multiple Of 3 Numbers

Introduction

Finding the least common multiple of 3 numbers is a fundamental skill in arithmetic that helps solve problems involving periodic events, fractions, and ratios. This guide explains step‑by‑step how to determine the least common multiple of 3 numbers using listing multiples, prime factorization, and the relationship with the greatest common divisor, ensuring clarity for beginners and efficiency for advanced learners It's one of those things that adds up..

Step‑by‑Step Method

Method 1: Listing Multiples

1. Write the first few multiples of each number separately.
   • For 4: 4, 8, 12, 16, 20, 24, 28, 32, …
   • For 6: 6, 12, 18, 24, 30, 36, 42, 48, …
   • For 9: 9, 18, 27, 36, 45, 54, 63, 72, …
2. Identify the smallest number that appears in all three lists.
   • In the example above, 24 is the first common entry, so the least common multiple of 4, 6, and 9 is 24.
3. Verify by checking that 24 is divisible by each original number without remainder.

Why it works: Listing multiples creates a visual set of common values; the smallest shared value is, by definition, the least common multiple.

Method 2: Prime Factorization

1. Factor each number into primes.
   • 4 = 2²
   • 6 = 2 × 3
   • 9 = 3²
2. Take the highest power of each prime that appears in any factorization.
   • Prime 2 appears as 2² (from 4).
   • Prime 3 appears as 3² (from 9).
   • Which means, LCM = 2² × 3² = 4 × 9 = 36.
3. Multiply these highest powers together to obtain the result.

Key point: The prime factorization method guarantees the correct LCM because it captures the maximum multiplicity of each prime factor needed for common divisibility.

Method 3: Using the Greatest Common Divisor (GCD)

The relationship between LCM and GCD for three numbers (a), (b), and (c) is:

[ \text{LCM}(a,b,c) = \frac{|a \times b \times c|}{\text{GCD}(a,b) \times \text{GCD}(ab, c)} ]

1. Find GCD(a, b).
   • GCD(4, 6) = 2.
2. Multiply the GCD by the remaining numbers to get an intermediate product:
   • 2 × 4 × 6 = 48.
3. Find GCD of that product with the third number (c).
   • GCD(48, 9) = 3.
4. Divide the product by this GCD:
   • 48 ÷ 3 = 16.
   • On the flip side, this result (16) is not the LCM; we must apply the formula correctly:

[ \text{LCM}(4,6,9) = \frac{4 \times 6 \times 9}{\text{GCD}(4,6) \times \text{GCD}(4 \times 6, 9)} = \frac{216}{2 \times 3} = \frac{216}{6} = 36 ]

Important: The GCD approach is efficient when you already know the GCDs, but it requires careful handling of intermediate products.

Scientific Explanation

The least common multiple of 3 numbers is the smallest positive integer that is divisible by each of the three numbers. Its existence is guaranteed because the set of common multiples is non‑empty (it includes the product of the three numbers) and can be ordered And that's really what it comes down to..

• Prime factorization works because any common multiple must contain each prime factor at least as many times as the highest exponent found in the individual factorizations.
• Listing multiples is a brute‑force method that directly reflects the definition but becomes impractical for large numbers.
• The GCD method leverages the fact that the product of two numbers equals the product of their GCD and LCM:

[ \text{LCM}(a,b) = \frac{|a \times b|}{\text{GCD}(a,b)} ]

Extending