Understanding the Context

The least common multiple of two integers is the smallest positive number that is evenly divisible by both. While listing multiples works, it becomes inefficient with larger numbers. Prime factorization offers a systematic and reliable approach based on the unique prime numbers that make up each factor.

Step-by-Step: Finding LCM(8, 12) Using Prime Factorizations

Step 1: Factor each number into primes

Begin by breaking down 8 and 12 into their prime components.

• 8:
  8 is an even number, so divide by 2 repeatedly:
  8 ÷ 2 = 4
  4 ÷ 2 = 2
  2 ÷ 2 = 1
  So, 8 = 2 × 2 × 2 = 2³

Key Insights

• 12:
  12 is also even:
  12 ÷ 2 = 6
  6 ÷ 2 = 3
  3 is prime
  So, 12 = 2 × 2 × 3 = 2² × 3¹

Step 2: List all prime factors with their highest powers

To find the LCM, take each prime number that appears in either factorization, and use the highest exponent that appears across the factorizations.

| Prime | Highest Power in 8 or 12 |
|-------|--------------------------|
| 2 | ³ (from 8 = 2³) |
| 3 | ¹ (from 12 = 3¹) |

Step 3: Multiply these primes raised to their highest powers

Now multiply them together:
LCM(8, 12) = 2³ × 3¹ = 8 × 3 = 24

Step 4: Verify the result

Check that 24 is divisible by both 8 and 12:

• 24 ÷ 8 = 3 ✔️
  
• 24 ÷ 12 = 2 ✔️

Final Thoughts

Yes, 24 is divisible by both, and since we used exponents optimally, it is the smallest such number.

Summary

To find the LCM of 8 and 12 using prime factorization:

1. Factor each number into primes:
   
   • 8 = 2³
     
   • 12 = 2² × 3¹
     
2. Take each prime with the highest exponent:
   
   • 2³ and 3¹
     
3. Multiply them together:
   
   • LCM = 2³ × 3 = 8 × 3 = 24

This method ensures accuracy and clarity, making it ideal not only for students learning LCM but also for teachers and math enthusiasts!

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