Question

The binary equivalent of 25 is

• A. (1100)\(_2\)
• B. (1001)\(_2\)
• C. (11001)\(_2\)
• D. (11101)\(_2\)

Hint:

When converting decimal to binary, quickly check your answer by converting it back. For (11001)\(_2\), the decimal value is \(1 \cdot 2^4 + 1 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0 = 16 + 8 + 0 + 0 + 1 = 25\). This confirms the result.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The question asks to convert a decimal number (base-10) to its binary equivalent (base-2). The standard method for this is the repeated division by 2.
Step 2: Key Formula or Approach:
To convert a decimal number to binary, we repeatedly divide the decimal number by 2 and record the remainders. The process continues until the quotient becomes 0. The binary number is then obtained by reading the remainders from the bottom up.
Step 3: Detailed Explanation:
Let's convert the decimal number 25 to binary:
\begin{itemize} \item Divide 25 by 2: Quotient = 12, Remainder = 1 \item Divide 12 by 2: Quotient = 6, Remainder = 0 \item Divide 6 by 2: Quotient = 3, Remainder = 0 \item Divide 3 by 2: Quotient = 1, Remainder = 1 \item Divide 1 by 2: Quotient = 0, Remainder = 1 \end{itemize} Reading the remainders from the bottom up gives us 11001.
So, \( (25)_{10} = (11001)_2 \).
Alternatively, we can express 25 as a sum of powers of 2:
The powers of 2 are ..., 32, 16, 8, 4, 2, 1.
25 can be written as:
\[ 25 = 16 + 8 + 1 \] \[ 25 = (1 \times 2^4) + (1 \times 2^3) + (0 \times 2^2) + (0 \times 2^1) + (1 \times 2^0) \] The coefficients of the powers of 2 give the binary representation: 11001.
Step 4: Final Answer:
The binary equivalent of 25 is (11001)\(_2\). This corresponds to option (C).