Ponder This Challenge - October 2024 - Splitting a number

This puzzle was suggested by Karl Heinz Hofmann - thanks!

We are looking for a natural number X satisfying the following properties of its decimal representation:

1. All the digits $\{1,2,3,4,6,7,8,9\}$ appear at least once, but $\{0, 5\}$ do not appear at all.

2. X can be written as a front part A and back part B, such that if X were a string, it could be written as A concatenated with B, such that

3. B is the product of the digits of X.

4. X is a multiple of B.

5. B is a perfect square.

For example, $X=3411296$ has the property that its digit product is $1296$ so it can be split into $A=341$ and $B=1296$ such that properties 2.1 and 2.3 are satisfied. However, none of the other properties are satisfied in this case since $X$ is missing the digits $7,8$ and is not a multiple of $B$.

Your goal: Find the least number X satisfying the above properties.

A bonus "*" will be given for finding the least number X satisfying the above properties except property 2.3 which is replaced by a new property: 2.3 The digit product of A is a perfect cube.

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Solution

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  October 2024 Solution:

  The numerical solutions are

  X = 1817198712146313216, A = 1817198712, B = 146313216
  X = 8419411123272236924928, A = 84194111232, B = 72236924928
  

  The solution is based on a search, which can be optimized in various ways. We give the method described by Alex Izvalov (thanks, Alex!):

  We'll use several tactics to reduce the field of search.- As the number $X$ should contain all the digits $1,2,3,4,6,7,8,9$ at least once, and $B$ is the product of digits of $X$, then $B = 2^{a+7}\cdot3^{b+4}\cdot7^{c+1}$ If $B$ is a perfect square, then it should be in the form: $B = 2^{2a+8}\cdot3^{2b+4}\cdot7^{2c+2}$, where $a,b,c$ are natural numbers

  • To search for $B$ in the (almost) increasing order we'll loop through the sum of $a,b,c, s = a+b+c$ and then we'll loop a from $0$ to $s, b$ from $0$ to $s-a$, and then $c = s-a-b$
  • Next we should check if $B$ contains digits $0$ or $5,$ and we'll proceed to the next cycle iteration in this case
  • As $X$ is a multiple of $B$, then (let $m$ be the length of $B$) $A\cdot 10^{m}+B$ is a multiple of $B$, then prime factoring of $A$ should have as many 3s and 7s as $B$ has, and the number of 2s should be not less than the number of 2s in $B$ minus the length of $B$. We'll call the minimal number consisting of these prime factors, $A_0$
  • As now we know the product of digits of $X$, and we know $B$, we'll know the product of digits of $A$, too (and for the advanced problem we can break the loop early if it's not a perfect cube).

  Then we can proceed in 2 directions: either search for $A$ as a multiple of $A_0\cdot q$ and loop through $q$ and then check the digits, or construct $A$ as a number of gradually increasing length which product of numbers should be equal to $\frac{B}{\text{prod}(B)}$, and then check if it's a multiple of $A_0$.

  I found the first, more straightforward way, easier to use when searching for the candidates for the minimal X, and the second way to prove that the results are indeed minimal.

Solvers

• *Bertram Felgenhauer (30/9/2024 4:40 PM IDT)
• *Lazar Ilic (30/9/2024 5:38 PM IDT)
• Paul Revenant (30/9/2024 6:07 PM IDT)
• *King Pig (30/9/2024 10:47 PM IDT)
• *Andreas Stiller (1/10/2024 3:21 AM IDT)
• Alex Fleischer (1/10/2024 8:47 AM IDT)
• Yi Jiang (1/10/2024 3:23 PM IDT)
• *Griffin Pinney (1/10/2024 4:09 PM IDT)
• *Guglielmo Sanchini (1/10/2024 5:17 PM IDT)
• John Tromp (1/10/2024 5:47 PM IDT)
• *Yan-Wu He (2/10/2024 4:36 AM IDT)
• *Robin Guilliou (2/10/2024 2:24 PM IDT)
• *Latchezar Christov (2/10/2024 3:07 PM IDT)
• *Daniel Bitin (2/10/2024 5:32 PM IDT)
• *Lawrence Au (2/10/2024 8:01 PM IDT)
• *Hugo Pfoertner (2/10/2024 8:31 PM IDT)
• Loukas Sidiropoulos (3/10/2024 3:02 AM IDT)
• *Sanandan Swaminathan (3/10/2024 2:10 PM IDT)
• *Daniel Chong Jyh Tar (3/10/2024 5:41 PM IDT)
• Reiner Martin (3/10/2024 10:52 PM IDT)
• *Maksym Voznyy (3/10/2024 11:21 PM IDT)
• *Lorenz Reichel (4/10/2024 5:44 PM IDT)
• Axel Amestoy (5/10/2024 3:48 PM IDT)
• *Harald Bögeholz (5/10/2024 7:38 PM IDT)
• William Han (5/10/2024 7:53 PM IDT)
• Rudy Cortembert (6/10/2024 11:29 AM IDT)
• *Alex Izvalov (6/10/2024 5:39 PM IDT)
• Guillaume Escamocher (6/10/2024 6:18 PM IDT)
• Sri Mallikarjun J (7/10/2024 5:26 AM IDT)
• *Juergen Koehl (7/10/2024 9:24 AM IDT)
• *Arthur Vause (7/10/2024 2:09 PM IDT)
• *Paul Lupascu (7/10/2024 4:20 PM IDT)
• Amos Guler (7/10/2024 6:34 PM IDT)
• *Radu-Alexandru Todor (8/10/2024 1:55 AM IDT)
• Stéphane Higueret (8/10/2024 4:36 PM IDT)
• *Martin Thorne (8/10/2024 9:01 PM IDT)
• Paul Shaw (8/10/2024 10:51 PM IDT)
• *Adrian Neacsu (8/10/2024 11:06 PM IDT)
• *Károly Csalogány (9/10/2024 12:03 PM IDT)
• *David Greer (10/10/2024 6:40 PM IDT)
• *Victor Chang (11/10/2024 12:37 AM IDT)
• Guangxi Liu (11/10/2024 7:26 AM IDT)
• *Ankit Aggarwal (11/10/2024 2:49 PM IDT)
• *Asaf Zimmerman (11/10/2024 5:01 PM IDT)
• Ashfaque Shaikh (11/10/2024 6:48 PM IDT)
• Piyush Agrawal (12/10/2024 6:04 AM IDT)
• Gary M. Gerken (12/10/2024 6:30 PM IDT)
• Jason Kim (12/10/2024 8:59 PM IDT)
• *Shouky Dan & Tamir Ganor (13/10/2024 9:28 AM IDT)
• Joaquim Carrapa (13/10/2024 1:06 PM IDT)
• *Stephen Ebert & Matthew Lee (15/10/2024 12:10 AM IDT)
• Terry Lee (15/10/2024 3:45 AM IDT)
• *Hadwin Jenkins (15/10/2024 11:37 AM IDT)
• Dieter Beckerle (16/10/2024 1:23 PM IDT)
• *Alper Halbutogullari (16/10/2024 1:32 PM IDT)
• Michael Liepelt (16/10/2024 9:54 PM IDT)
• Diana Gornea (17/10/2024 2:33 PM IDT)
• *Vladimir Volevich (18/10/2024 2:18 PM IDT)
• Yunkyu (James) Lee (19/10/2024 5:04 PM IDT)
• Suyeong Hahn (19/10/2024 7:53 PM IDT)
• Divy Soni (19/10/2024 8:09 PM IDT)
• *Marco Bellocchi (20/10/2024 1:53 AM IDT)
• Shirish Chinchalkar (21/10/2024 6:41 AM IDT)
• Michael Oosterhagen (21/10/2024 11:26 AM IDT)
• *Kang Jin Cho (21/10/2024 7:54 PM IDT)
• *Erik Wünstel (21/10/2024 10:40 PM IDT)
• *Mihai Stancu (22/10/2024 1:49 PM IDT)
• *Nyles Heise (22/10/2024 11:07 PM IDT)
• Benjamin Buhrow (23/10/2024 1:04 AM IDT)
• John Spurgeon and Kipp Johnson (23/10/2024 5:41 AM IDT)
• Stefan Wirth (26/10/2024 4:44 AM IDT)
• Ken Dsouza (27/10/2024 10:48 AM IDT)
• Frank Schoeps (27/10/2024 3:15 PM IDT)
• *Franciraldo Cavalcante (28/10/2024 7:41 AM IDT)
• *Dan Dima (28/10/2024 12:13 PM IDT)
• *Justin Snopek (29/10/2024 12:38 AM IDT)
• Benjamin Cha (29/10/2024 1:05 AM IDT)
• *Motty Porat (29/10/2024 3:00 PM IDT)
• *Davide Colì (30/10/2024 2:07 AM IDT)
• Evan Semet (30/10/2024 6:25 AM IDT)
• Ido Meisner (30/10/2024 5:53 PM IDT)
• Harshit Kothari (31/10/2024 1:09 AM IDT)
• Tim Walters (31/10/2024 12:06 PM IDT)
• *David F.H. Dunkley (31/10/2024 12:24 PM IDT)
• Alfred Reich (31/10/2024 6:20 PM IDT)
• Chris Shannon (31/10/2024 7:46 PM IDT)
• Maths ForFun2 (31/10/2024 11:57 PM IDT)
• Blaine Hill (1/11/2024 9:01 PM IDT)
• *Andrea Andenna (2/11/2024 1:20 PM IDT)
• Julian Ma (2/11/2024 2:27 PM IDT)
• Oscar Volpatti (2/11/2024 8:14 PM IDT)