To find the binary equivalent of the decimal number 13, we will convert the decimal number into binary using the method of successive division by 2. Let's go through the steps in detail. ### Step-by-Step Conversion from Decimal to Binary 1. **Understanding Binary System**: - The binary system is a base-2 numeral system that uses only two digits: 0 and 1. Each digit represents a power of 2, starting from \(2^0\) on the right. 2. **Division by 2**: - To convert a decimal number to binary, we repeatedly divide the number by 2 and keep track of the remainders. The binary equivalent is formed by reading the remainders in reverse order (from bottom to top). 3. **Performing the Division**: - Start with the decimal number 13. - Divide 13 by 2: - \(13 \div 2 = 6\) with a remainder of **1**. - Next, take the quotient (6) and divide by 2: - \(6 \div 2 = 3\) with a remainder of **0**. - Take the next quotient (3) and divide by 2: - \(3 \div 2 = 1\) with a remainder of **1**. - Finally, take the last quotient (1) and divide by 2: - \(1 \div 2 = 0\) with a remainder of **1**. 4. **Collecting the Remainders**: - Now, we collect the remainders from the last division to the first: - The remainders are: 1 (from 1), 1 (from 3), 0 (from 6), 1 (from 13). - Reading these from bottom to top gives us: **1101**. 5. **Final Result**: - Therefore, the binary equivalent of the decimal number 13 is **1101**. ### Why Option A is Correct - The correct binary representation of the decimal number 13 is **1101**, which matches option A. ### Why Other Options are Incorrect - **Option B: 1011** - This binary number represents \(1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 8 + 0 + 2 + 1 = 11\). Thus, it is not equal to 13. - **Option C: 1110** - This binary number represents \(1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 8 + 4 + 2 + 0 = 14\). Therefore, it is also not equal to 13. - **Option D: 1001** - This binary number represents \(1 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 8 + 0 + 0 + 1 = 9\). Hence, it is not equal to 13. ### Summary of Key Points - The binary equivalent of a decimal number is found by dividing the number by 2 and recording the remainders. - The correct binary representation of 13 is **1101** (Option A). - Other options (B, C, D) do not represent the decimal number 13 when converted back to decimal. - Always read the remainders from bottom to top to get the correct binary number. This method of conversion is fundamental in understanding how different numeral systems work, especially in computer science and digital electronics.