Question

The decimal number $25_{10}$ is represented in binary as _ _ _ _ .}

• A. 1 1 0 0 1
• B. 1 0 1 0 1
• C. 1 1 1 0 0
• D. 1 0 0 1 1

Hint:

To verify a binary number quickly, multiply each bit by its corresponding power of 2 and add the results. If the sum equals the original decimal number, the conversion is correct.

Answer 1

The Correct Option is A

Concept: To convert a decimal number into binary, we repeatedly divide the number by 2 and record the remainders. The binary representation is obtained by reading the remainders from bottom to top.

Step 1: Perform repeated division by 2. \[ 25 \div 2 = 12 \] Remainder: \[ 1 \] \[ 12 \div 2 = 6 \] Remainder: \[ 0 \] \[ 6 \div 2 = 3 \] Remainder: \[ 0 \] \[ 3 \div 2 = 1 \] Remainder: \[ 1 \] \[ 1 \div 2 = 0 \] Remainder: \[ 1 \]

Step 2: Write remainders in reverse order. Reading from bottom to top: \[ 1\quad1\quad0\quad0\quad1 \] Therefore, \[ 25_{10}=11001_2 \]

Step 3: Verification using powers of 2. \[ 11001_2 \] \[ =(1\times2^4)+(1\times2^3)+(0\times2^2)+(0\times2^1)+(1\times2^0) \] \[ =16+8+0+0+1 \] \[ =25 \] Hence, the conversion is correct. \[ \boxed{11001_2} \]