If 7-digit number 485A64B is divisible by 8 and 9, then find the greatest value of A+ B? a)18             b) 6        c) 9             d) 12

To solve this problem, we must use the divisibility rules for both 8 and 9.

### Step 1: Divisibility Rule for 8
A number is divisible by 8 if its last three digits are divisible by 8. In this case, the last three digits are \( A64B \).

Since \( A64B \) represents a 4-digit number where \( A \) and \( B \) are digits, we need to ensure that the last three digits, \( 64B \), form a number divisible by 8.

Now, let's check possible values of \( B \) by forming numbers of the form \( 64B \):
- If \( B = 0 \), then \( 640 \div 8 = 80 \) (divisible by 8).
- If \( B = 1 \), then \( 641 \div 8 = 80.125 \) (not divisible by 8).
- If \( B = 2 \), then \( 642 \div 8 = 80.25 \) (not divisible by 8).
- If \( B = 3 \), then \( 643 \div 8 = 80.375 \) (not divisible by 8).
- If \( B = 4 \), then \( 644 \div 8 = 80.5 \) (not divisible by 8).
- If \( B = 5 \), then \( 645 \div 8 = 80.625 \) (not divisible by 8).
- If \( B = 6 \), then \( 646 \div 8 = 80.75 \) (not divisible by 8).
- If \( B = 7 \), then \( 647 \div 8 = 80.875 \) (not divisible by 8).
- If \( B = 8 \), then \( 648 \div 8 = 81 \) (divisible by 8).
- If \( B = 9 \), then \( 649 \div 8 = 81.125 \) (not divisible by 8).

Therefore, the possible values for \( B \) based on divisibility by 8 are \( B = 0 \) or \( B = 8 \).

### Step 2: Divisibility Rule for 9
A number is divisible by 9 if the sum of its digits is divisible by 9. The given number is \( 485A64B \), and the sum of its digits is:
\[
4 + 8 + 5 + A + 6 + 4 + B = 27 + A + B
\]
For divisibility by 9, \( 27 + A + B \) must be divisible by 9. Since 27 is already divisible by 9, \( A + B \) must also be divisible by 9.

### Step 3: Combining the Conditions
From the divisibility by 8, we have two possibilities for \( B \): \( B = 0 \) or \( B = 8 \).

- If \( B = 0 \), then \( A + 0 \) must be divisible by 9. The only value for \( A \) that satisfies this condition is \( A = 9 \).
- If \( B = 8 \), then \( A + 8 \) must be divisible by 9. The only value for \( A \) that satisfies this condition is \( A = 1 \).

### Step 4: Maximizing \( A + B \)
We are asked to find the greatest value of \( A + B \). The two possibilities are:
- \( A = 9 \) and \( B = 0 \), so \( A + B = 9 \).
- \( A = 1 \) and \( B = 8 \), so \( A + B = 9 \).

In both cases, \( A + B = 9 \).

Thus, the greatest value of \( A + B \) is \( \boxed{9} \).

### Additional Questions:
1. Can you explain the divisibility rule for 8 in more detail?
2. What other numbers follow both the divisibility rules for 8 and 9?
3. How would the solution change if the number was divisible by 6 instead of 9?
4. Why is the sum of digits used for checking divisibility by 9?
5. What happens if \( A + B \) doesn't meet the divisibility requirement?

### Tip:
Always check the divisibility of smaller parts of a number first (like the last three digits for divisibility by 8) before considering the entire number.

If 7-digit number 485A64B is divisible by 8 and 9, then find the greatest value of A + B? a) 18 b) 6 c) 9 d) 12

Solve the problem of finding the greatest value of A + B for the 7-digit number 485A64B, using divisibility rules for 8 and 9. Learn how the last three digits and sum of digits affect divisibility in this arithmetic problem. Suitable for Grades 6-8.

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