Which digit should come in place of * and $ if the number 62684*$ is divisible by both 8 and 5?

Options

A. 4, 0
B. 0, 4
C. 0, 0
D. 4, 4

Answer

To solve the problem of finding which digits should replace * and $ in the number 62684*$ so that it is divisible by both 8 and 5, we can follow these steps:

### Step 1: Understand the divisibility rules
- A number is divisible by **5** if its last digit (unit digit) is either **0** or **5**.
- A number is divisible by **8** if the number formed by its last three digits is divisible by **8**.

### Step 2: Determine the last digit ($)
Since the last digit must make the number divisible by 5, we can conclude that $ must be either **0** or **5**.

### Step 3: Check the implications of $ being 0 or 5
- If $ = 0, then the last three digits of the number become **4*S*0**.
- If $ = 5, then the last three digits become **4*S*5**.

### Step 4: Check divisibility by 8 for both cases
1. **Case 1: $ = 0**
- The last three digits are **4S0**.
- For divisibility by 8, we check if 4S0 is divisible by 8.
- The possible values for S are 0 to 9, so we can check:
- If S = 0, then 400 ÷ 8 = 50 (divisible)
- If S = 1, then 410 ÷ 8 = 51.25 (not divisible)
- If S = 2, then 420 ÷ 8 = 52.5 (not divisible)
- If S = 3, then 430 ÷ 8 = 53.75 (not divisible)
- If S = 4, then 440 ÷ 8 = 55 (divisible)
- If S = 5, then 450 ÷ 8 = 56.25 (not divisible)
- If S = 6, then 460 ÷ 8 = 57.5 (not divisible)
- If S = 7, then 470 ÷ 8 = 58.75 (not divisible)
- If S = 8, then 480 ÷ 8 = 60 (divisible)
- If S = 9, then 490 ÷ 8 = 61.25 (not divisible)
- Valid values for S when $ = 0 are **0, 4, and 8**.

2. **Case 2: $ = 5**
- The last three digits are **4S5**.
- For divisibility by 8, we check if 4S5 is divisible by 8.
- The possible values for S are again 0 to 9:
- If S = 0, then 405 ÷ 8 = 50.625 (not divisible)
- If S = 1, then 415 ÷ 8 = 51.875 (not divisible)
- If S = 2, then 425 ÷ 8 = 53.125 (not divisible)
- If S = 3, then 435 ÷ 8 = 54.375 (not divisible)
- If S = 4, then 445 ÷ 8 = 55.625 (not divisible)
- If S = 5, then 455 ÷ 8 = 56.875 (not divisible)
- If S = 6, then 465 ÷ 8 = 58.125 (not divisible)
- If S = 7, then 475 ÷ 8 = 59.375 (not divisible)
- If S = 8, then 485 ÷ 8 = 60.625 (not divisible)
- If S = 9, then 495 ÷ 8 = 61.875 (not divisible)
- There are **no valid values for S** when $ = 5.

### Step 5: Conclusion
The only valid combinations for (S, $) that make the number 62684*$ divisible by both 8 and 5 are:
- (0, 0)
- (4, 0)
- (8, 0)

### Final Answer
Thus, the digits that can replace * and $ are:
- **0, 0** or **4, 0** or **8, 0**.