The maximum three digit integer in the decimal system will be represented in the binary system by which one of the following?

Options

A. `(1111110001)_(2)`
B. `(111111110)_(2)`
C. `(1111100111)_(2)`
D. `(1111000111)_(2)`

Answer

To find the maximum three-digit integer in the decimal system and its representation in the binary system, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the maximum three-digit integer in the decimal system.**
- The maximum three-digit integer is **999**.

2. **Convert the decimal number 999 into binary.**
- To convert a decimal number to binary, we can use the method of successive division by 2 and recording the remainders.
- Start with the number 999 and divide it by 2, keeping track of the quotient and the remainder.

| Division by 2 | Quotient | Remainder |
|---------------|----------|-----------|
| 999 ÷ 2 | 499 | 1 |
| 499 ÷ 2 | 249 | 1 |
| 249 ÷ 2 | 124 | 1 |
| 124 ÷ 2 | 62 | 0 |
| 62 ÷ 2 | 31 | 0 |
| 31 ÷ 2 | 15 | 1 |
| 15 ÷ 2 | 7 | 1 |
| 7 ÷ 2 | 3 | 1 |
| 3 ÷ 2 | 1 | 1 |
| 1 ÷ 2 | 0 | 1 |

3. **Read the binary number from the bottom up.**
- The binary representation of 999 is obtained by reading the remainders from bottom to top: **1111100111**.

4. **Verify the binary representation.**
- To ensure that the conversion is correct, we can convert the binary number back to decimal:
- \(1 \times 2^9 + 1 \times 2^8 + 1 \times 2^7 + 1 \times 2^6 + 1 \times 2^5 + 0 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0\)
- \(= 512 + 256 + 128 + 64 + 32 + 0 + 0 + 4 + 2 + 1 = 999\)

5. **Conclusion:**
- The binary representation of the maximum three-digit integer (999) in the decimal system is **1111100111**.