to find the smallest value of n

Question

Find the smallest natural number n such that there exist infinitely many solutions to [a1,a2...,an]/a1+a2+...+an= k , for all k ∈ N. Here [a , b]=LCM(a , b). I tried it and it seems very odd as we need to find p/q=k[p and q having no common factor] and this works for every K belonging to N. Can anyone help me how this is possible?

Answer 1

Note $$\frac{[1,2k,2k+1]}{1+2k+(2k+1)}=\frac{2k(2k+1)}{4k+2}=k, $$ so $n=3$ is certainly big enough. Can you see why $n<3$ does not work for all $k$?