Le Monde puzzle [#849]

A straightforward Le Monde mathematical puzzle:

Find a pair (a,b) of integers such that a has an odd number d of digits larger than 2 and ab is written as 10^d+1+10a+1. Find the smallest possible values of a and of b.

I ran the following R code

d=3
for (a in 10^(d-1):(10^d-1)){
  c=10^(d+1)+10*a+1
  if (a*trunc(c/a)==c)
       print(c(a,c))}

which produced a=137 (and b=83) as the unique case. For d=4, I obtained a=9091 and b=21, for d=6, a=909091, and b=21, for d=7, a=5882353 and b=27, while for d=5, my code did not return any solution. While d=8 took too long to run, a prime factor decomposition of 10⁹+1 leads to (with the schoolmath R library)

> for (d in 3:10) print(c(d,prime.factor(10^(d+1)+1)))
[1]   3  73 137
[1]    4   11 9091
[1]    5  101 9901
[1]      6     11 909091
[1]       7      17 5882353
[1]     8     7    11    13    19 52579
[1]     9   101  3541 27961
[1]   10   11   11   23 4093 8779

which gives a=52631579 and b=29 for d=8 and also explains why there is no solution for d=5. The corresponding a has too many digits!

This issue of Le Monde Science&Médecine leaflet had more interesting entries, from one on “LaTeX as the lingua franca of mathematicians”—which presumably made little sense to any reader unfamiliar with LaTeX—to the use of “big data” tools (like news rover) to analyse data produce by the medias, to  yet another tribune of Marco Zito about the “five sigma” rule used in particle physics (and for the Higgs boson analysis)—with the reasonable comment that a large number of repetitions of an experiment is likely to exhibit unlikely events, and an also reasonable recommendation to support “reproduction experiments” that aim at repeating exceptional phenomena—, to a solution to puzzle #848—where the resolution is the same as mine’s, but mentions the principle of Dirichlet’s drawers to exclude the fact that all prices are different, a principle I had never heard off…