Le Monde puzzle [#1134]

A weekly Monde current mathematical puzzle on gcd’s and scm’s:

If one replaces a pair (a,b) of integers with the pair (g,s) of their greatest common denominator and smallest common multiple, how long at most before the sequence ends. Same question when considering a collection of five integers where two are selected by the pair (g,s) of their greatest common denominator and smallest common multiple.

The first question is straightforward as s is a multiple of s. So the sequence ends at most after one run. For five, run of a brute force R search return 9 as “the” solution (even though the true maximum is 10, as illustrated by the quintuplet (16,24,36,54,81):

ogcd <- function(x,y){r<-x%%y
  return(ifelse(r,ogcd(y,r),y))}

oscm<-function(x,y) x*y/ogcd(x,y)

divemul<-function(a,b) return(c(oscm(a,b),ogcd(a,b)))

for (t in 1:1e5){
ini=sample(1:1e2,5)
i=0;per=ker=sample(ini,2)
nez=divemul(per[1],per[2])
while(!max(nez%in%per)){
 ini=c(ini[!ini%in%per],nez)
 per=sample(ini,2)
 ker=rbind(ker,per)
 nez=divemul(per[1],per[2])
 i=i+1}
 sol=max(sol,i)}