To solve this problem, we need to find the minimum value of $AB$, where the 12-digit number "20251034B15A" is divisible by 21.

Step 1: Divisibility Rule for 21

The number is divisible by 21 if it is divisible by both 3 and 7 (since $21 = 3 \times 7$).

Step 2: Divisibility by 3

A number is divisible by 3 if the sum of its digits is divisible by 3.

Given number: $20251034B15A$

• Known digits sum: $2 + 0 + 2 + 5 + 1 + 0 + 3 + 4 + 1 + 5 = 23$
• To ensure divisibility by 3, we need $23 + B + A \equiv 0 \pmod{3}$.

Thus, $23 + B + A \equiv 0 \pmod{3}$, or equivalently, $B + A \equiv 1 \pmod{3}$.

Step 3: Divisibility by 7

Divisibility by 7 is more complex and often involves trial