Puzzle
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Ponder This Challenge - June 2020 - Farewell and welcome to Gadi Aleksandrowicz!

This is the 278th Ponder This monthly challenge. If you sum the divisors of 278 (excluding itself), you get 142, which is the number of challenges I wrote for IBM's puzzle corner. There are 138 natural numbers that are smaller than 278 and relatively prime to it. 138 is also the number of challenges our best solver to date, Daniel Bitin, has correctly solved.

It was an honor serving as IBM research's PuzzleMaster for more than a decade. As I'm leaving IBM, I welcome Gadi Aleksandrowicz, our new PuzzleMaster, and devote this month's challenge to him.

Your challenge is to find a number whose sum of divisors (excluding itself) is 12,142,680,281,284,711,468,101,282,998,309,016,699,980,172 and exactly 3,031,634,148,236,289,733,373,855,928,919,180,891,127,808 out of the natural numbers that are smaller than it are relatively prime to it.

A bonus '*' for finding the connection to Gadi.

Solution

  • The solution is 9,099,305,265,787,740,660,101,533,112,532,340,984,145,268. This number can be found by finding the preimage of the Euler totient function on 3,031,634,148,236,289,733,373,855,928,919,180,891,127,808, and computing the sum of divisors for each element of the preimage. All this can be accomplished relatively easily given the prime factorization of the numbers involved. See "Euler's Totient Function and Its Inverse" by Hansraj Gupta for additional details.

    This number is Oded's way of welcoming me, as it turns out this is the ASCII encoding of “https://gadial.net”, the link to my (Hebrew) math blog. Thanks, Oded, and farewell!

Solvers

  • *Bert Dobbelaere (31/05/2020 04:44 PM IDT)
  • *Anders Kaseorg (01/06/2020 03:30 AM IDT)
  • *Li Li (01/06/2020 07:38 AM IDT)
  • *Bertram Felgenhauer (01/06/2020 04:37 PM IDT)
  • Alper Halbutogullari (02/06/2020 03:37 PM IDT)
  • *Oscar Volpatti (03/06/2020 02:12 AM IDT)
  • Guillermo Wildschut (03/06/2020 11:16 AM IDT)
  • *Dominik Reichl (03/06/2020 06:07 PM IDT)
  • *Uoti Urpala (03/06/2020 09:22 PM IDT)
  • *Lawrence Au (04/06/2020 09:28 AM IDT)
  • Max A. Alekseyev (04/06/2020 05:39 PM IDT)
  • *David Greer (05/06/2020 07:01 PM IDT)
  • Arthur Vause (05/06/2020 09:08 PM IDT)
  • *Florian Fischer (07/06/2020 02:50 AM IDT)
  • *Eden Saig (6/6/2020 04:46 PM IDT)
  • *Jim and Kevin Roche (08/06/2020 11:39 PM IDT)
  • *Willem Hendriks (09/06/2020 09:52 AM IDT)
  • *Andreas Stiller (09/06/2020 12:37 PM IDT)
  • Sean Egan (11/06/2020 02:08 AM IDT)
  • Hugo Pfoertner (11/6/2020 09:17 PM IDT)
  • Keith Schneider (12/6/2020 02:58 AM IDT)
  • Vladimir Volevich (12/6/2020 10:08 PM IDT)
  • *Latchezar Christov (13/6/2020 10:23 PM IDT)
  • *Haye Chan (14/6/2020 03:21 AM IDT)
  • Walter Sebastian Gisler (14/6/2020 08:03 AM IDT)
  • Daniel Bitin (14/6/2020 05:57 PM IDT)
  • Lorenz Reichel (22/6/2020 02:00 PM IDT)
  • *Todd Will (23/6/2020 01:59 AM IDT)
  • Joerg Schepers (23/6/2020 11:11 AM IDT)
  • *Gary M. Gerken (27/6/2020 06:49 AM IDT)
  • Daniel Chong Jyh Tar (28/6/2020 01:50 PM IDT)
  • *Kang Jin Cho (29/6/2020 07:35 PM IDT)
  • *Motty Porat (29/6/2020 01:21 PM IDT)
  • Radu-Alexandru Todor (30/6/2020 06:51 PM IDT)
  • *Zhou Hangbo (30/6/2020 08:19 PM IDT)
  • JJ Rabeyrin (30/6/2020 09:51 PM IDT)

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