Determine how many 1000 digit numbers A have the following property:
When any digit of A, aside from the first, is deleted to form a 999 digit number B, then B divides A.
To determine the number of 1000-digit numbers A with the given property, we can break down the problem into smaller steps.
Step 1: Find the number of choices for the first digit of A.
Since A is a 1000-digit number, the first digit cannot be zero. Therefore, there are 9 choices (1-9) for the first digit.
Step 2: Find the number of choices for the remaining 999 digits of A.
For each of the remaining 999 digits, it can be any digit from 0 to 9. Therefore, there are 10 choices for each digit.
Step 3: Determine which of the 999 digits of A can be deleted to form B.
For any digit of A (aside from the first digit), it can be deleted to form B if and only if B divides A. In other words, the digit must be a divisor of A. To check if a digit is a divisor of A, we can divide A by the digit and check if the remainder is zero.
Step 4: Count the number of digits that can be deleted for each A.
For each of the 999 digits of A, we check if it is a divisor of A. If it is a divisor, then it can be deleted, and B divides A. We count the number of digits that satisfy this condition.
Step 5: Calculate the total number of 1000-digit numbers A.
To find the total number of 1000-digit numbers A with the given property, we multiply the number of choices for each step. The result is the product of the choices in step 1 (9 choices for the first digit) and step 2 (10 choices for each of the remaining 999 digits). Finally, we multiply this result by the number of digits that can be deleted (counted in step 4).
By following these steps, we can determine the total number of 1000-digit numbers A that satisfy the given property.