# 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.

**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.

### Your Answer

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