# Le Monde puzzle [#1134]

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**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)}

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