To solve the problem, we need to convert the number ( 26_{10} ) (which is in decimal) to its binary equivalent ( X_{2} ) (which is in base 2). Let's go through the steps to find the correct binary representation of the decimal number 26. Step-by-Step Conversion from Decimal to Binary

1. Understanding the Decimal System: The decimal system (base 10) uses digits from 0 to 9. Each digit's position represents a power of 10. For example, the number 26 can be broken down as: [ 2 \times 10^1 + 6 \times 10^0 = 20 + 6 = 26 ]
2. Understanding the Binary System: The binary system (base 2) uses only two digits: 0 and 1. Each position represents a power of 2. For example, the binary number ( 11010_2 ) can be broken down as: [ 1 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 16 + 8 + 0 + 2 + 0 = 26 ]
3. Conversion Process:
4. Start with the decimal number 26.
5. Divide the number by 2 and keep track of the quotient and the remainder.
6. The remainder will give you the binary digits from least significant to most significant.

Here’s how the division works: - ( 26 \div 2 = 13 ) remainder ( 0 ) - ( 13 \div 2 = 6 ) remainder ( 1 ) - ( 6 \div 2 = 3 ) remainder ( 0 ) - ( 3 \div 2 = 1 ) remainder ( 1 ) - ( 1 \div 2 = 0 ) remainder ( 1 )

1. Collecting the Remainders: Now, we collect the remainders from the last division to the first:
2. The remainders collected in reverse order are ( 11010 ).
3. Final Result: Therefore, the binary representation of ( 26_{10} ) is ( 11010_2 ).

Conclusion The correct answer is B. 11010. Explanation of Other Options

• A. 11011: This binary number equals ( 1 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 16 + 8 + 0 + 2 + 1 = 27_{10} ). This is greater than 26, so it is incorrect.
• C. 11110: This binary number equals ( 1 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 16 + 8 + 4 + 2 + 0 = 30_{10} ). This is also greater than 26, so it is incorrect.
• D. 10111: This binary number equals ( 1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 16 + 0 + 4 + 2 + 1 = 23_{10} ). This is less than 26, so it is incorrect.

Revision Summary

• To convert a decimal number to binary, repeatedly divide by 2 and record the remainders.
• Collect the remainders in reverse order to get the binary representation.
• The binary representation of ( 26_{10} ) is ( 11010_2 ).
• Always verify by converting the binary back to decimal to ensure accuracy.