By replacing each of a, b with the appropriate digit from 0 to 9, find all numbers of the form 8726a2415b, which are exactly divisible by 9.

To determine which numbers of the form \( 8726a2415b \) are divisible by 9, we use the rule that a number is divisible by 9 if the sum of its digits is divisible by 9.

First, let's express the sum of the digits:

\[ 8 + 7 + 2 + 6 + a + 2 + 4 + 1 + 5 + b \]

Calculating the sum of the known digits:

\[ 8 + 7 + 2 + 6 + 2 + 4 + 1 + 5 = 35 \]

Now we include \( a \) and \( b \):

\[ 35 + a + b \]

For the number to be divisible by 9, the total sum \( 35 + a + b \) must be divisible by 9.

Since \( a \) and \( b \) are digits from 0 to 9, we need to find values of \( a \) and \( b \) such that:

\[ 35 + a + b \equiv 0 \pmod{9} \]

First, calculate \( 35 \mod 9 \):

\[ 35 \div 9 = 3 \text{ remainder } 8 \]

So,

\[ 35 \equiv 8 \pmod{9} \]

We need:

\[ 8 + a + b \equiv 0 \pmod{9} \]

Which simplifies to:

\[ a + b \equiv 1 \pmod{9} \]

Next, we find pairs of \( a \) and \( b \) that satisfy this condition:

\[
\begin{align*}
a = 0, & \quad b = 1 \\
a = 1, & \quad b = 0 \\
a = 2, & \quad b = 8 \\
a = 3, & \quad b = 7 \\
a = 4, & \quad b = 6 \\
a = 5, & \quad b = 5 \\
a = 6, & \quad b = 4 \\
a = 7, & \quad b = 3 \\
a = 8, & \quad b = 2 \\
a = 9, & \quad b = 1 \\
\end{align*}
\]

So, the pairs \((a, b)\) that make \( 8726a2415b \) divisible by 9 are:

- \( (0, 1) \)
- \( (1, 0) \)
- \( (2, 8) \)
- \( (3, 7) \)
- \( (4, 6) \)
- \( (5, 5) \)
- \( (6, 4) \)
- \( (7, 3) \)
- \( (8, 2) \)
- \( (9, 1) \)

Each pair provides a different number \( 8726a2415b \) that is divisible by 9.

Would you like more details or have any questions?

Here are 8 related questions for further practice:

1. What is the rule for a number to be divisible by 3?
2. Find all numbers of the form \( 1234x \) that are divisible by 9.
3. How many digits are there in the number \( 99999 \times 9 \)?
4. What are the factors of 81?
5. If \( y = 5 \) and \( z = 4 \), find a number of the form \( 7yz8 \) that is divisible by 9.
6. Find the sum of the digits of the number \( 987654321 \).
7. How many pairs of digits (a, b) satisfy \( a + b \equiv 2 \pmod{9} \)?
8. Prove that any rearrangement of the digits of 123456789 is divisible by 9.

**Tip:** Always double-check your results by substituting values back into the original equation or rule to ensure correctness.

Learn how to find numbers of the form 8726a2415b that are divisible by 9. Explore the rule of divisibility by 9 and find suitable values for digits a and b to satisfy the condition. Suitable for middle school students learning about number theory and divisibility rules.

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