Question

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. \(12\)

Answer 1

The Correct Option is C

To find the greatest value of \(A + B\) such that the 7-digit number 485A64B is divisible by both 8 and 9, we follow these steps:

Step 1: Divisibility by 8

A number is divisible by 8 if its last three digits are divisible by 8. Here, the last three digits are 64B. We need 64B to be divisible by 8. Let's test values of \(B: 0 \leq B \leq 9\).

Checking divisibility for each \(B\):

• B = 0: 640/8 = 80 (divisible)
• B = 1: 641/8 = 80.125 (not divisible)
• B = 2: 642/8 = 80.25 (not divisible)
• B = 3: 643/8 = 80.375 (not divisible)
• B = 4: 644/8 = 80.5 (not divisible)
• B = 5: 645/8 = 80.625 (not divisible)
• B = 6: 646/8 = 80.75 (not divisible)
• B = 7: 647/8 = 80.875 (not divisible)
• B = 8: 648/8 = 81 (divisible)
• B = 9: 649/8 = 81.125 (not divisible)

The possible values of \(B\) that make 64B divisible by 8 are 0 and 8.

Step 2: Divisibility by 9

A number is divisible by 9 if the sum of its digits is divisible by 9. The sum of the digits in 485A64B is \(4 + 8 + 5 + A + 6 + 4 + B = 27 + A + B\).

Check divisibility for each \(B\) that works from Step 1:

• B = 0: 27 + A + 0 must be divisible by 9.
• B = 8: 27 + A + 8 = 35 + A must be divisible by 9.

For B = 0: \(27 + A\) must be 36, 45, etc., so possible \(A\) = 9. Then, \(A + B = 9 + 0 = 9.\)

For B = 8: \(35 + A\) must be 36, 45, etc., so possible \(A\) = 1. Then, \(A + B = 1 + 8 = 9.\)

Thus, the greatest value of \(A + B\) that satisfies both conditions is \(9\).

The correct answer is \(9\).