Can you find the unique ten digit number

There is a unique number of ten digits, for which the following holds-

  • all digits from 0 up to 9 occur exactly once in the number;
  • the first digit is divisible by 1;
  • the number formed by the first two digits is divisible by 2;
  • the number formed by the first three digits is divisible by 3;
  • the number formed by the first four digits is divisible by 4;
  • the number formed by the first five digits is divisible by 5;
  • the number formed by the first six digits is divisible by 6;
  • the number formed by the first seven digits is divisible by 7;
  • the number formed by the first eight digits is divisible by 8;
  • the number formed by the first nine digits is divisible by 9;
  • the number formed by the ten digits is divisible by 10.

 

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amol Scholar Asked on 15th August 2015 in Math Puzzles.
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  • 1 Answer(s)

    The requested number is 3816547290.

    Tenth digit
    A number is divisible by 10 if it ends on a 0. Therefore, the tenth digit of the requested number must be a 0.

    Fifth digit
    A number is divisible by 5 if it ends on a 0 or 5. The 0 has already been used, so the fifth digit of the requested number is a 5.

    First digit
    A number is always divisible by 1. Nothing can be said about the first digit.

    Second digit
    A number is divisible by 2 if it is even, so if it ends on a 0, 2, 4, 6, or 8. The 0 has already been used, so the second digit of the requested number is a 2, 4, 6, or 8.

    The fourth, sixth, and eighth digits of the requested number must also be divisible by two, so these digits must be 2, 4, 6, or 8 too. The digits on the first, third, fifth, seventh, and ninth positions of the requested number can only be 1, 3, 5, 7, or 9.

    Third digit
    A number is divisible by 3 if the sum of its digits is divisible by 3. Below all possibilities for the first three digits of the requested number (first and third digits are 1, 3, 5, 7, or 9, second digit is 2, 4, 6, or 8, and the sum of the digits is divisible by 3):

        123    723    147    183    783
    129 729 741 189 789
    321 921 369 381 981
    327 927 963 387 987

    Fourth digit
    A number is divisible by 4 if:

      • the number ends on a 0, 4, or 8 and the last-but-one digit is even, or

     

      • the number ends on a 2 or 6 and the last-but-one digit is odd.

     

    The third digit of the requested number is odd, so the fourth digit can only be a 2 or 6. Below are all possibilities for the first four digits of the requested number:

        1236    9216    3692    3812    7892
    1296 9276 9632 3816 7896
    3216 1472 1832 3872 9812
    3276 1476 1836 3876 9816
    7236 7412 1892 7832 9872
    7296 7416 1896 7836 9876

    Sixth digit
    A number is divisible by 6 if it is divisible by 2 and 3, so if it ends on a 0, 2, 4, 6, or 8, and the sum of the digits is divisible by 3. The first three digits of the requested number are already divisible by 3, so the sum of the fourth, fifth, and sixth digits must be divisible by 3 too. Below are the two possibilities for the fourth, fifth, and sixth digits of the requested number (fourth digit is 2, or 6, fifth digit is 5, sixth digit is 2, 4, 6, or 8, and the sum of the digits is divisible by 3):

        258    654

    Combined with what we already know about the first five digits, this gives the following possibilities for the first sixth digits of the requested number:

        123654    723654    147258    183654    783654
    129654 729654 741258 189654 789654
    321654 921654 369258 381654 981654
    327654 927654 963258 387654 987654

    Eighth digit
    A number is divisible by 8 if:

      • the number formed by the last two digits is divisible by 8 and the last-but-two digit is even, or

     

      • the number formed by the last two digits minus 4 is divisible by 8 and the last-but-two digit is odd.

     

    The last-but-two digit is the sixth digit of the requested number, and is a 4 or 8. Therefore, the number formed by the seventh and eighth digits must be divisible by 8. In addition, we know that the seventh digit must be odd. These are the possible combinations:

        16    32    56    72    96

    Combined with what we already know about the first six digits, this gives the following possibilities for the first eight digits of the requested number:

        18365472    74125896
    18965432 78965432
    18965472 98165432
    38165472 98165472
    14725896 98765432

    Seventh digit
    The number formed by the first seven digits of the requested number must be divisible by 7. For the numbers shown above, this only holds for the number 38165472.

    Ninth digit
    For the ninth digit, only the digit 9 remains. Note that every number formed by the digits 1 up to 9 is divisible by 9. A number is divisible by 9 if the sum of its digits is divisible by 9. The sum of the digits 1 up to 9 is 45, which is divisible by 9.

    Conclusion
    The requested number is 3816547290.

    amol Scholar Answered on 15th August 2015.
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