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.