## 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:

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

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?

• benvitalis says:

Ooops! I forgot to type the sign divide ” | “

• benvitalis says:

Thanks for mentioning it.

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