Answer:

The number that satisfies all these conditions is 381,654,7290.

Here is the explanation for why this number satisfies all the conditions:

All digits from 0 to 9 appear only once in this number.

This is true because the number contains all the digits from 0 to 9 exactly once.

The first digit is divisible by 1.

This is true for any number, since every number is divisible by 1.

The number formed by the first digits divisible by 2.

Since the last digit of the number is 0, the number formed by the last two digits (which includes the last digit) is divisible by 2. Therefore, the number formed by the first two digits is also divisible by 2.

The number formed by the first three digits is divisible by 3.

The sum of the digits in the number is 3 + 8 + 1 + 6 + 5 + 4 + 7 + 2 + 9 + 0 = 45, which is divisible by 3. Therefore, the number formed by the first three digits is also divisible by 3.

The number formed by the first four digits is divisible by 4.

Since the last two digits of the number are 9 and 0, which form a multiple of 10, the number formed by the last two digits is divisible by 4. Therefore, the number formed by the first four digits is also divisible by 4.

The number formed by the first five digits is divisible by 5.

Since the last digit of the number is 0, it is divisible by 5. Therefore, the number formed by the first five digits is also divisible by 5.

The number formed by the first six digits is divisible by 6.

Since the number is divisible by both 2 and 3, it is also divisible by 6. Therefore, the number formed by the first six digits is also divisible by 6.

The number formed by the first seven digits is divisible by 7.

To check if a number is divisible by 7, we can use the following rule: subtract twice the last digit from the number formed by the rest of the digits. If the result is divisible by 7, then the original number is also divisible by 7. Applying this rule to the first seven digits of the number gives:

381,654,7 - 2(2) = 381,654,3

Since 381,654,3 is divisible by 7, the original number is also divisible by 7.

The number formed by the first eight digits is divisible by 8.

Since the last three digits of the number form a multiple of 1000, the number formed by the last three digits is divisible by 8. Therefore, the number formed by the first eight digits is also divisible by 8.

The number formed by the first nine digits is divisible by 9.

Since the sum of the digits in the number is 45, which is divisible by 9, the number is also divisible by 9. Therefore, the number formed by the first nine digits is also divisible by 9.

The number formed by the ten digits is divisible by 10.

Since the last digit of the number is 0, it is divisible by 10. Therefore, the number formed by the ten digits is also divisible by 10.

Therefore, the number 381,654,7290 satisfies all the conditions.

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