To determine the possible values of $a$ such that the number $148101a095$ is divisible by 11, we can use the divisibility rule for 11:

Divisibility Rule for 11: A number is divisible by 11 if the difference between the sum of the digits in odd positions and the sum of the digits in even positions is either 0 or a multiple of 11.

Step-by-Step Solution:

1. Identify the positions of the digits in $148101a095$:

   • Odd positions: 1, 8, 0, $a$, 9
   • Even positions: 4, 1, 1, 0, 5

2. Calculate the sum of digits in odd positions: $1 + 8 + 0 + a + 9 = 18 + a$

3. Calculate the sum of digits in even positions: $4 + 1 + 1 + 0 + 5 = 11$

4. Find the difference between the sums: $(18 + a) - 11 = a + 7$

5. Set up the condition for divisibility by 11: $a + 7 \equiv 0 \pmod{11}$ This implies: $a + 7 = 11k \quad \text{(for some integer } k)$ $a = 11k - 7$

6. Find the valid digit $a$ (since $a$ must be a digit, $a$ is an integer from 0 to 9): $a = 11k - 7$

   If $k = 1$: $a = 11 \times 1 - 7 = 4$ $k = 2$ gives $a = 22 - 7 = 15$, which is not a valid digit.

Conclusion:

The only valid digit for $a$ is $4$.

So, the possible value of $a$ is $\boxed{4}$.

Would you like further details or have any questions? Here are some related questions you might find interesting:

1. How does the divisibility rule for other numbers like 7 or 13 work?
2. What happens if the digit $a$ is placed in different positions of the number?
3. How can the rule of divisibility by 11 be extended to larger numbers?
4. How would the approach change if the number were in a different base system, like base 7 or base 12?
5. What if $a$ were a negative digit, how would that affect the solution?

Tip: Always double-check the conditions for divisibility, as they can help simplify the problem significantly!