Question

The binary representation of the decimal number 125 is

• A. 1110001
• B. 1010101
• C. 1111101
• D. 1111001

Hint:

When converting from decimal to binary, repeatedly divide by 2 and use the remainders to form the binary number.

Answer 1

The Correct Option is C

Step 1: The smallest power of 2 at or above 125 is \( 2^7=128 \), and \( 128-125=3 \).
Step 2: Write 128 in binary as \( 10000000 \), and 3 in binary as \( 0000011 \).
Step 3: Subtracting, \( 10000000-0000011=1111101 \), which is the binary form of 125.
\[ \boxed{125_{10} = 1111101_2} \]

Answer 2

Since this is a multiple-choice question, another reliable approach is to convert each proposed binary string back into decimal directly and see which one equals 125.

1. 1110001: Reading place values from left to right as \( 64,32,16,8,4,2,1 \): \( 1\times64+1\times32+1\times16+0\times8+0\times4+0\times2+1\times1=64+32+16+0+0+0+1=113 \). Not 125.
2. 1010101: \( 1\times64+0\times32+1\times16+0\times8+1\times4+0\times2+1\times1=64+16+4+1=85 \). Not 125.
3. 1111101: \( 1\times64+1\times32+1\times16+1\times8+1\times4+0\times2+1\times1=64+32+16+8+4+0+1=125 \). Matches exactly.
4. 1111001: \( 1\times64+1\times32+1\times16+1\times8+0\times4+0\times2+1\times1=64+32+16+8+0+0+1=121 \). Not 125.

Working backward through place values shows only one of the four binary strings actually sums to 125.

Therefore, the correct answer is 1111101.