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: Find the largest multiple of \( 16^2=256 \) that fits in \( 2000 \): \( 7\times256=1792 \), leaving a remainder of \( 2000-1792=208 \).
Step 2: Find the largest multiple of \( 16 \) that fits in \( 208 \): \( 13\times16=208 \), leaving a remainder of \( 0 \); \( 13 \) is written as \( D \) in hex.
Step 3: The digits found are \( 7 \), \( D \), \( 0 \), read from the highest place value down.
\[ \boxed{2000_{10} = 7D0_{16}} \]

Answer 2

Since this is a multiple-choice question, we can also work backward: convert each proposed hexadecimal answer directly to decimal using place values \( 16^2=256 \), \( 16^1=16 \), \( 16^0=1 \), and see which one equals \( 2000 \).

1. 7C8: \( C=12 \). \( 7\times256+12\times16+8\times1=1792+192+8=1992 \), \( 8 \) short of \( 2000 \).
2. 7D0: \( D=13 \). \( 7\times256+13\times16+0\times1=1792+208+0=2000 \), matches exactly.
3. 7A6: \( A=10 \). \( 7\times256+10\times16+6\times1=1792+160+6=1958 \), \( 42 \) short of \( 2000 \).
4. 7B4: \( B=11 \). \( 7\times256+11\times16+4\times1=1792+176+4=1972 \), \( 28 \) short of \( 2000 \).

Working backward from the options, only \( 7D0 \) reconstructs to exactly \( 2000 \) in decimal.

Therefore, the correct answer is 7D0.