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LCM and GCD fraction (Posted on 2025-05-14)

Find the least possible value for the fraction

lcm(a,b)+lcm(b,c)+lcm(c,a)
--------------------------------
gcd(a,b)+gcd(b,c)+gcd(c,a)

over all distinct positive integers a, b, c.

No Solution Yet Submitted by Danish Ahmed Khan

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computer exploration

| Comment 1 of 2

The answer seems to be 2.5 from:

min=9999; clc

mina=0;minb=0;minc=0;

for tot=6:200

for a=1:tot/3

for b=a+1:(tot-a)/2

c=tot-a-b;

if c>b

v=(lcm(a,b)+lcm(a,c)+lcm(c,b))/(gcd(a,b)+gcd(a,c)+gcd(c,b));

if v<=min

fprintf('%16.13f %7d %7d %7d',[min mina minb minc])

fprintf('\n')

min=v;

mina=a;

minb=b;

minc=c;

end

settles in to multiples of:

     fraction          a       b       c

 2.5000000000000       1       2       4

Posted by Charlie on 2025-05-14 15:42:03

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