What is the minimum possible LCM of $X$ natural numbers whose sum is $N$?
I faced a specific problem in my module with $X=10$ and $N=45$ and I solved it by semi-brute-force. Any ideas on how to solve the general problem?
[Math] the minimum possible LCM of $X$ natural numbers whose sum is $N$
elementary-number-theory
Best Answer
Answer seems to jump around a lot as you vary the parameters. Let's take $X=10$ and $40\le N\le50$. Writing $a^b$ for "$b$ copies of $a$" I get $$\matrix{N&a_i&{\rm LCM}\cr40&4^{10}&4\cr41&6^621^3&6\cr42&5^81^2&5\cr43&6^6321^2&6\cr44&6^62^4&6\cr45&6^71^3&6\cr46&5^91&5\cr47&6^72^21&6\cr48&6^72^3&6\cr49&6^732^2&6\cr50&5^{10}&5\cr}$$ I prefer the term "educated trial and error" to the term "semi-brute force", but it's your dime.