To solve the problem, we need to determine which of the given statements are necessary to identify a three-digit number where each digit is different from the other.
### Step-by-Step Solution:
1. **Understanding the Problem**: We need to find a three-digit number where each digit is unique. The digits can range from 0 to 9, but since it’s a three-digit number, the hundreds place cannot be 0.
2. **Analyzing Statement I**:
- Statement I states that each of the digits of the given number is a multiple of 3.
- The possible digits that are multiples of 3 from 0 to 9 are: 0, 3, 6, and 9.
- However, since it is a three-digit number, we cannot use 0 in the hundreds place.
- Thus, the possible digits are 3, 6, and 9.
3. **Analyzing Statement II**:
- Statement II states that the digit in the unit's place is 50% less than that in the hundred's place.
- Let’s denote the digit in the hundreds place as X. According to the statement, the unit's place digit would be X - (50% of X) = X - (X/2) = X/2.
- Since X must be a multiple of 3 (from Statement I), we can check the possible values:
- If X = 3, then the unit's place would be 3/2 = 1.5 (not valid).
- If X = 6, then the unit's place would be 6/2 = 3 (valid).
- If X = 9, then the unit's place would be 9/2 = 4.5 (not valid).
- Thus, if X = 6, the unit's place is 3. The remaining digit for the tens place must be 9 (since all digits must be different).
4. **Analyzing Statement III**:
- Statement III states that none of the digits is zero.
- Since we already established that the digits we are considering (3, 6, and 9) do not include 0, this statement does not add any new information.
5. **Conclusion**:
- From Statement I, we identified the possible digits as 3, 6, and 9.
- From Statement II, we determined the specific arrangement of these digits to form the number 693.
- Statement III is not necessary since it does not affect the outcome.
### Final Answer:
The necessary statements to answer the question are:
- Statement I and Statement II.
