Digit puzzle – 9 and 10 distinct digits

 
 
(1)

Find all integers   N   whose decimal representation

N \; = \; abcdefghi

consists of 9 distinct nonzero digits such that

2 \; | \; (a - b)
3 \; | \; (a - b + c)
4 \; | \; (a - b + c - d)
5 \; | \; (a - b + c - d + e)
6 \; | \; (a - b + c - d + e - f)
7 \; | \; (a - b + c - d + e - f + g)
8 \; | \; (a - b + c - d + e - f + g - h)
9 \; | \; (a - b + c - d + e - f + g - h + i)

 

(2)

Do the same for integers   N \; = \; abcdefghij

consisting of 10 distinct digits (leading zeros excluded) such that:

  2 \; | \; (a - b)
  3 \; | \; (a - b + c)
  4 \; | \; (a - b + c - d)
  5 \; | \; (a - b + c - d + e)
  6 \; | \; (a - b + c - d + e - f)
  7 \; | \; (a - b + c - d + e - f + g)
  8 \; | \; (a - b + c - d + e - f + g - h)
  9 \; | \; (a - b + c - d + e - f + g - h + i)
10 \; | \; (a - b + c - d + e - f + g - h + i - j)

 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged , . Bookmark the permalink.

5 Responses to Digit puzzle – 9 and 10 distinct digits

  1. pipo says:

    Ben, what do you mean here?

  2. pipo says:

    The 9 digit-number is: 958473261
    The 10-digit-number: according to me not possible because you can not put a zero in a 10-digitnumber fitting the rules.

    pipo

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