Suppose it was 5, 7, 8: LCM = 280 — no. - Richter Guitar

Understanding the Context

The LCM of two or more integers is the smallest positive integer divisible by each number without leaving a remainder. For multiple numbers, it’s found by combining their prime factorizations and taking the highest power of each prime.

For example:

• Prime factors:
  
  • 5 = 5
    
  • 7 = 7
    
  • 8 = 2³

LCM = 2³ × 5 × 7 = 8 × 5 × 7 = 280 — but only if considering 5, 7, and 8 together. However, this example gets tricky when applied to any three numbers because pairing them differently affects the outcome.

Why 5, 7, and 8 Do Not Have LCM = 280 (in All Contexts)

Key Insights

At first glance, 5, 7, and 8 appear to multiply cleanly: 5 × 7 × 8 = 280. While this product is close, it’s not the LCM — and here’s why:

• LCM requires the shared least multiple, not the full product.
  
• The actual LCM of 5, 7, and 8 depends on their prime factors:
  
  • 5 = 5
    
  • 7 = 7
    
  • 8 = 2³
    → LCM = 2³ × 5 × 7 = 280 only if 5, 7, and 8 are all the numbers, and there’s no smaller common multiple than 280.
    But this false friendship between total product (280) and actual smallest multiple makes it tempting — yet mathematically flawed.

Real Example: When Do 5, 7, and 8 Actually Yield LCM = 280?

Only if you compute:
LCM(5, 7) = 35
Then LCM(35, 8) = 35 × 8 / GCD(35,8) = 280, since GCD(35,8) = 1.
So:
LCM(5, 7, 8) = 280 only when properly computed, which confirms the value, but the claim that “5, 7, 8 always = 280” is misleading due to confusing total product with LCM.

Common Mistake: Assuming Product Equals LCM

Final Thoughts

Many learners mistakenly believe that multiplying numbers gives the LCM. While 5×7×8 = 280, the LCM uses distinct prime factors, so multiplying counts repeated and unnecessary factors. The correct method avoids double-counting; LCM relies on highest exponents.

Practical Tips to Work LCM Correctly

• Use prime factorization for accuracy, especially with three or more numbers.
  
• Avoid assuming total product = LCM; test with smaller multiples.
  
• When dealing with sets like 5, 7, 8, focus on shared divisibility rather than multiplication.
  
• Apply LCM rules in real scenarios: scheduling events, dividing resources, or syncing cycles in math problems.

Conclusion

While 5, 7, and 8 produce a product of 280, this does not make their LCM 280 — at least not in general mathematical terms. The true LCM of 5, 7, and 8 is 280, but this arises from precise factorization, not simple multiplication. Always compute LCM by identifying the smallest number divisible by all inputs, not by multiplying them. Clear understanding of LCM prevents errors and builds confidence in solving number theory problems.

Key Takeaways:

• LCM ≤ product, but not always equal.
  
• Factorization reveals true shared multiples.
  
• Misconceptions come from confusing total product with LCM definition.