a ten-digit integer whose digits are all distinct. the number formed by the first n of them is divisible by n for each n from 1 to 10. what's the number?

Question

Its a 10 digit number.

Answer 1

 

Since it is a 10 digit number, and the digits ar distinct, the digits in the numbers are 0, 1, 2, 3, 4, 5 ,6 , 7, 8, and 9 

 

Taking advantage of the divisibility shortcut rules ...

 

Since  the first n numbers is a number divisible by n ...

 

    The 10 digit number is divisible by 10 ... so the last digiti is 0.

 

        xxxxxxxxx0

 

    The  5 digit number  must be a number divisible by 5 ... so the 5th digit must be 0 0r 5.  We have already used 0 and the digits are distinct.  So the 5th digit is 5.

 

        xxxx5xxxx0

 

    The remaining even digits ... 2, 4 , 6, 8 ... must each reside in an even numbered slot ... 2, 4, 6. or 8

 

    Also. the sum of the first 3 digits and the sum of the first 6 digits has to be a number divisible by 3 ... because both those numbers have to be divisible by 3

 

     The 3rd and 4th digit must represent a number divisible by 4. 

 

     The sum of the first 9 digits must be a number divisible by 9

 

From here ... it is a trial and error process until you end up with a number that meets the criteria specified. 

 

     

The answer is:

       

    3816547290

     

CHECK:

 

                        3 ÷ 1 = 3

                      38 ÷ 2 = 19

                    381 ÷ 3 = 127

                  3816 ÷ 4 = 954

                38165 ÷ 5 = 7633

              381654 ÷ 6 = 63609

            3816547 ÷ 7 = 545221

          38165472 ÷ 8 = 4770684

        381654729 ÷ 9 = 42406081

      3816547290 ÷ 10 = 381654729