We begin by testing a configuration where the numbers are distinct primes or mostly coprime small integers. However, since the sum is fixed at 140 and we have 7 numbers, we cannot use 7 large primes (their sum would likely exceed 140). We consider using small numbers that are pairwise coprime and whose LCM is large. - Groen Casting

Understanding the Context

Why Focus on Small Coprime Integers?

Distinct prime numbers are naturally coprime, but large primes quickly surpass the fixed sum. For example, the 7 smallest primes—2, 3, 5, 7, 11, 13, 17—sum to only 58. While smaller and coprime, this total is far below 140. Including larger primes like 29, 31, or others pushes the sum beyond 140, especially across seven terms. Thus, a smarter strategy involves using small base numbers that are pairwise coprime yet permit a meaningful total—without breaching 140.

Maximizing Coprimality Without Large Primes

Rather than relying solely on primes, consider small composite and composite-like integers that are pairwise coprime. These numbers avoid common factors and help reach a larger total within constraints. Pairwise coprimality ensures no shared prime factors across all selected numbers.

Key Insights

For instance, numbers like 2, 3, 5, 7, 11 (the first seven primes) remain fundamental but sum too low. To reach 140, we need a strategy that increases the total through coprime composites or modified entries—while preserving simplicity and avoids redundancy.

Key Insight: Use Small Coprime Building Blocks

A practical configuration begins with the smallest pairwise coprime set: often 2, 3, 5, and selected odd composites rooted in disjoint prime factors—e.g., 7 (prime), 10 (2×5), but 10 shares prime factors with 2 and 5. Instead, favor numbers derived from new or unused primes, such as 13, 17, 19—adding these increases the total.

For example:

• Start with 2, 3, 5, 7, 11 — sum so far: 28
  
• Add 13, 17 — sum so far: 50
  
• Remaining five numbers must total 90, filled with distinct integers coprime to all previously chosen and to each other

Final Thoughts

To keep numbers small yet large enough, choose small numbers like:

• 1 (coprime with all),
  
• 9 = 3² but 3 already used → avoid for coprimality,
  
• 25 = 5² → avoid (shared factor),
  
• 49 = 7² → avoid,
  
• So alternatives: 7 × 1 = 7 (taken), so try 121? Too large.

Instead, favorable coprime small integers below 50: 1, 9, 25, 49 (but coprimality fails due to prime factors shared). A better tactic: use numbers like:

• 1 (coprime with all),
  
• 4 = 2² → avoid due to 2,
  
• 9 = 3² → avoid,
  
• 25 = 5² → avoid,
  
• 49 = 7² → avoid,
  
• 121 too big,
  
• So choose: 1, 25, 49, but these conflict with prime factors—unless excluded.

Alternatively, relax strict primes and embrace small composites with disjoint prime supports:

• 1 (special unit),
  
• 9 (3²),
  
• 25 (5²),
  
• 49 (7²),
  
• 121 too big,
  
• But can’t include two squares sharing primes.