If the number 556743A57B is divisible by 8 and 9, then find the value of A-B.

Options

A. A)0
B. B)3
C. C)2
D. D)1

Answer

To solve the problem, we need to determine the values of A and B in the number 556743A57B such that the entire number is divisible by both 8 and 9.

### Step 1: Check divisibility by 8
For a number to be divisible by 8, the last three digits of the number must be divisible by 8. In this case, the last three digits are 57B.

To find the value of B, we will check the numbers formed by replacing B with digits from 0 to 9 and see which of them make 57B divisible by 8.

- 570 ÷ 8 = 71.25 (not divisible)
- 571 ÷ 8 = 71.375 (not divisible)
- 572 ÷ 8 = 71.5 (not divisible)
- 573 ÷ 8 = 71.625 (not divisible)
- 574 ÷ 8 = 71.75 (not divisible)
- 575 ÷ 8 = 71.875 (not divisible)
- 576 ÷ 8 = 72 (divisible)
- 577 ÷ 8 = 72.125 (not divisible)
- 578 ÷ 8 = 72.25 (not divisible)
- 579 ÷ 8 = 72.375 (not divisible)

From this, we find that B must be 6 for 57B to be divisible by 8.

### Step 2: Check divisibility by 9
Next, we need to ensure that the entire number is divisible by 9. A number is divisible by 9 if the sum of its digits is divisible by 9.

The digits of the number 556743A576 are:
5 + 5 + 6 + 7 + 4 + 3 + A + 5 + 7 + 6

Calculating the sum:
= 5 + 5 + 6 + 7 + 4 + 3 + A + 5 + 7 + 6
= 43 + A

Now, we need to find A such that 43 + A is divisible by 9.

Calculating the remainder of 43 when divided by 9:
43 ÷ 9 = 4 remainder 7.

To make 43 + A divisible by 9, A must be such that:
(7 + A) is divisible by 9.

The smallest non-negative value for A that satisfies this is:
A = 2 (since 7 + 2 = 9, which is divisible by 9).

### Step 3: Find A - B
Now we have:
A = 2
B = 6

We need to find A - B:
A - B = 2 - 6 = -4.

### Final Answer
The value of A - B is -4.

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