Question

The hexadecimal representation of the decimal number 2000 is

• A. 7C8
• B. 7D0
• C. 7A6
• D. 7B4

Hint:

When converting from decimal to hexadecimal, repeatedly divide by 16 and use the remainders to form the hexadecimal number.

Answer 1

The Correct Option is B

Step 1: Conversion from decimal to hexadecimal.
To convert the decimal number 2000 to hexadecimal, we repeatedly divide the number by 16 and record the remainders: \[ 2000 \div 16 = 125 \quad \text{remainder} \, 0 \] \[ 125 \div 16 = 7 \quad \text{remainder} \, 13 \, (\text{which is} \, D) \] \[ 7 \div 16 = 0 \quad \text{remainder} \, 7 \] Thus, the hexadecimal representation of 2000 is \( 7D0 \).
Step 2: Conclusion.
Therefore, the correct answer is (2) 7D0.

Answer 2

Instead of dividing by 16 repeatedly, we can convert 2000 to binary first and group the binary digits into sets of four to read off the hexadecimal digits directly, then check this against each option.

\( 2000 = 1024+512+256+128+64+16 = 2^{10}+2^9+2^8+2^7+2^6+2^4 \), giving the binary string \( 11111010000 \). Grouping into 4-bit nibbles from the right: \( 0111\ 1101\ 0000 \). Converting each nibble: \( 0111=7 \), \( 1101=13=D \), \( 0000=0 \). So the hexadecimal representation is \( 7D0 \).

1. 7C8: Converting back: \( 7\times256 + 12\times16 + 8 = 1792+192+8=1992 \), not \( 2000 \).
2. 7D0: Converting back: \( 7\times256+13\times16+0=1792+208+0=2000 \), matches exactly.
3. 7A6: Converting back: \( 7\times256+10\times16+6=1792+160+6=1958 \), not \( 2000 \).
4. 7B4: Converting back: \( 7\times256+11\times16+4=1792+176+4=1972 \), not \( 2000 \).

Only \( 7D0 \) reconstructs to exactly \( 2000 \) when converted back to decimal, confirming both the binary-grouping derivation and the option check agree.

Therefore, the correct answer is 7D0.