# Can you find the unique ten digit number

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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.

amol Scholar Asked on 15th August 2015 in

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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