A nine-digit number is formed using each of the digits 1,2,3,...,9 exactly once. For n = 1,2,3,...,9, n divides the first n digits of the number. Find the number.
To find the number, we need to start by understanding the problem requirements.
The requirements state that for each digit n from 1 to 9, the first n digits of the number should be divisible by n. This means that the number must be formed in such a way that each successive portion of digits from the left is divisible by the corresponding digit n.
Let's analyze the situation step by step:
Step 1: The first digit should be divisible by 1. Since there is only one digit remaining, the first digit must be 1.
Step 2: The first two digits should be divisible by 2. This means that the second digit must be even. The only even digit available is 2, so the first two digits must be 12.
Step 3: The first three digits should be divisible by 3. The sum of the digits in 12 is 1 + 2 = 3, which is divisible by 3. So, the first three digits are fine.
Step 4: The first four digits should be divisible by 4. The last two digits of 12 are 1 and 2, which together form the number 12. Since 12 is divisible by 4, the first four digits are fine.
Step 5: The first five digits should be divisible by 5. The last digit of 12 is not divisible by 5 since it is 2. Therefore, we need to rearrange the remaining digits 3, 4, 5, 6, 7, 8, and 9 to form a number that is divisible by 5. The only option available is to place 5 as the second digit, resulting in 152.
Step 6: The first six digits should be divisible by 6. The last two digits of 152 are 5 and 2, which together form the number 52. Since 52 is not divisible by 6, we need to rearrange the remaining digits to form a number that is divisible by 6. The only option available is to place 6 as the last digit, resulting in 1562.
Step 7: The first seven digits should be divisible by 7. To check this, we need to examine the first seven digits of 1562. There is no specific rule for divisibility by 7 with multiple digits, so we would have to try different combinations of the remaining digits to see if any form a number divisible by 7.
By trying different combinations, we find that combining the remaining digits, 3, 4, 7, 8, and 9, to form the number 37984 gives us a number divisible by 7. So, the first seven digits are fine.
Step 8: The first eight digits should be divisible by 8. The last three digits of 37984 are 9, 8, and 4, which together form the number 984. Since 984 is divisible by 8 (since 8 divides 984 evenly), the first eight digits are fine.
Step 9: The first nine digits should be divisible by 9. To check this, we need to examine the entire number, 3798491562. The sum of the digits in this number is 3 + 7 + 9 + 8 + 4 + 9 + 1 + 5 + 6 + 2 = 54, which is divisible by 9. So, the first nine digits meet the requirements.
Therefore, the number formed using each of the digits 1, 2, 3, ..., 9 exactly once, and satisfying the given conditions is 3798491562.