Problems & Puzzles: Puzzles

Puzzle 1158 A unique polydivisible integer
Shyam Sunder Gupta pulished in the chapter 3 of his book  “Creative Puzzles
 to Ignite Your Mind”, his puzzle 57

"Find a ten-digit number that uses all the digits 0 to 9 exactly once such that the first n-digit number is divisible by n for all values of n from 1 to 10 i.e., the first 2-digit number is divisible by 2, the first 3-digit number is divisible by 3 and so on."

Q. Send your solution.

During the week 13-19 Jan 2024, contributions came from Michael Branicky, Gennady Gusev, JM Rebert, Emmanuel Vantieghem, Giorgos Kalogeropoulos, Jeff Heleen, Metin Sariyar, Paul Cleary, Paolo Lava, Ken Wilke, Adam Stinchcombe, Oscar Volaptti

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All of them found the unique solution:3816547290.

With more or less the same argument when they sent it. The following is the argument from Volpatti

A solution must satisfy many constraints.

 
Divisibility by 2
Digits in even positions must be even.
Hence odd digits must be in odd positions.

 
Divisibility by 5
 
Tenth digit must be 0, fifth digit must be 5.

 
Divisibility by 4
 
Fourth digit and eighth digit must be congruent to 2 mod 4, so they must be 2 and 6, in some order.
Hence digits 4 and 8 must be placed in second and sixth positions, in some order.

 
Divisibility by 8
If eighth digit is 2*d, then seventh digit is congruent to -d mod 4.

 
Divisibility by 3
The sums of the following three-digit blocks is divisible by 3:
first to third,
 
fourth to sixth,
seventh to ninth.

 
Only ten candidates satisfy them all:

 
1472589630
1836547290
1896543270
1896547230
3816547290
7412589630
7896543210
9816543270
9816547230
9876543210

Seventh number is divisible by 7 only for candidate 3816547290.

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On Jan 22, 2024 Dmitry Kamenetsky wrote:

Puzzle 1158 is a known problem

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