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We are tasked with finding the minimum value of \( AB \) in the 12-digit number \( 20251034B15A \), which is divisible by 21.

### Step 1: Divisibility by 21
A number divisible by 21 must be divisible by both 3 and 7. We will need to apply the rules of divisibility for both these numbers.

#### Divisibility by 3:
A number is divisible by 3 if the sum of its digits is divisible by 3. Let's calculate the sum of the known digits:

\[
2 + 0 + 2 + 5 + 1 + 0 + 3 + 4 + 1 + 5 = 23
\]

Let \( B \) and \( A \) be the digits to be determined. The total sum of the digits will be:

\[
23 + B + A
\]

For divisibility by 3, we need:

\[
23 + B + A \equiv 0 \pmod{3}
\]

Thus,

\[
(B + A) \equiv 1 \pmod{3}
\]

This equation will guide us when selecting values for \( A \) and \( B \).

#### Divisibility by 7:
To check divisibility by 7, we use the complete number. We will try different values for \( B \) and \( A \), making sure both divisibility rules are satisfied.

### Step 2: Minimize \( AB \)
We want to minimize \( AB \) while ensuring the number is divisible by both 3 and 7. We will test combinations of \( A \) and \( B \) that satisfy the modulo conditions.

I'll compute this next to determine the minimal values.The possible values of \( A \) and \( B \) that satisfy the divisibility rules (by both 3 and 7) are:

- \( A = 1 \), \( B = 0 \) with \( AB = 10 \)
- \( A = 2 \), \( B = 8 \) with \( AB = 28 \)
- \( A = 4 \), \( B = 3 \) with \( AB = 43 \)
- \( A = 7 \), \( B = 6 \) with \( AB = 76 \)
- \( A = 9 \), \( B = 1 \) with \( AB = 91 \)

The minimum value of \( AB \) is **10**, which occurs when \( A = 1 \) and \( B = 0 \).

Would you like more details on how divisibility by 7 was checked, or have any other questions?

### Related Questions:
1. What is the rule for divisibility by 7, and how can it be applied quickly?
2. How does the sum of digits help in checking divisibility by 3?
3. Can \( AB \) have any other values for different divisibility conditions?
4. How does changing one digit in a large number affect divisibility?
5. What if the condition was divisibility by another number, like 11 or 13?

### Tip:
When checking divisibility by 7, you can break large numbers into smaller chunks and apply modular arithmetic to simplify the process.

Given that 20251034B15A is a 12-digit number divisible by 21, find the minimum value of AB.

Learn how to find the minimum value of AB in the 12-digit number 20251034B15A, divisible by 21. Explore divisibility rules for 3 and 7, modular arithmetic, and number theory in this problem suitable for Grades 9-12.

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