Xi'an's Og

Another low-key arithmetic problem as Le Monde current mathematical puzzle:

Notice that there are 10 numbers less than, and prime with 11, 100 less than and prime with 101, 1000 less than, and prime with 1111? What is the smallest integer N such that the cardinal of the set of M<N prime with N is 10⁴? What is the smallest multiple of 1111 using only different digits? And what is the largest?

As it is indeed a case for brute force resolution as in the following R code

library(numbers)
homanycoprim=function(n){
  many=1
  for (i in 2:(n-1)) many=many+coprime(i,n)
  return(many)}

smallest=function(){
  n=1e4
  many=homanycoprim(n)
  while (many!=1e4) many=homanycoprim(n<-n+1)
  return(n)}

which returns n=10291 as the smallest value of N.  For the other two questions, the usual integer2digit function is necessary

smalldiff=function(){
  n=1111;mul=1
  while (mul<1e6) {
    x=as.numeric(strsplit(as.character(n*mul), "")[[1]])
    while (sum(duplicated(x))!=0){
     mul=mul+1
     x=as.numeric(strsplit(as.character(n*mul), "")[[1]])}
  print(n*mul);mul=mul+1}}

leading to 241,087 as the smallest and 9,875,612,340 as the largest (with 10 digits).

This entry was posted on September 18, 2019 at 12:19 am and is filed under R, Statistics with tags brute-force solution, digits, intToDigits(), Le Monde, mathematical puzzle, R, strsplit(). You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.