Select the correct option: Convert the decimal number 101 (base 10) into its binary representation.

Introduction / Context:
This question is from number systems and digital representation. You are asked to convert the decimal number 101 (base 10) into binary. Understanding how to convert between decimal and binary is important in computer science, digital electronics and aptitude tests involving number bases. The binary system is base 2, using only the digits 0 and 1, so we must express 101 as a sum of powers of 2 and then write the corresponding binary digits.

Given Data / Assumptions:

- The given number is 101 in decimal (base 10). - We must convert this number into base 2, that is, binary. - Binary digits (bits) are 0 and 1 only. - We are to select the correct binary representation from the options.

Concept / Approach:
The standard method of converting a decimal number to binary is repeated division by 2, keeping track of remainders. The binary digits are obtained by reading the remainders from last to first. Another viewpoint is to find which powers of 2 sum up to the decimal number, and then mark 1s in those positions and 0s elsewhere. For a number like 101, both methods are manageable and lead to the same binary representation. We will use the repeated division method here for clarity.

Step-by-Step Solution:
Step 1: Start with the decimal number 101. Step 2: Divide 101 by 2. The quotient is 50 and the remainder is 1. This remainder is the least significant bit (rightmost bit). Step 3: Divide the quotient 50 by 2. The new quotient is 25 and the remainder is 0. Step 4: Divide 25 by 2. The quotient is 12 and the remainder is 1. Step 5: Divide 12 by 2. The quotient is 6 and the remainder is 0. Step 6: Divide 6 by 2. The quotient is 3 and the remainder is 0. Step 7: Divide 3 by 2. The quotient is 1 and the remainder is 1. Step 8: Divide 1 by 2. The quotient is 0 and the remainder is 1. We stop when the quotient becomes 0. Step 9: Write the remainders from last to first: 1 1 0 0 1 0 1. Step 10: Hence, the binary representation of 101 is 1100101.

Verification / Alternative check:
We can verify by converting 1100101 back to decimal. From left to right, the bits correspond to 2^6, 2^5, 2^4, 2^3, 2^2, 2^1 and 2^0. The places with 1s are 2^6, 2^5, 2^2 and 2^0. Compute: 2^6 = 64, 2^5 = 32, 2^2 = 4 and 2^0 = 1. Summing these gives 64 + 32 + 4 + 1 = 101, which matches the original decimal number. This confirms that 1100101 is correct.

Why Other Options Are Wrong:
Option A (1101001) corresponds to 2^6 + 2^5 + 2^3 + 2^0 = 64 + 32 + 8 + 1 = 105. Option B (1100111) represents 64 + 32 + 4 + 2 + 1 = 103. Option C (1101011) represents 64 + 32 + 8 + 2 + 1 = 107. None of these equal 101 in decimal. Only option D (1100101) evaluates exactly to 101 and is therefore correct.

Common Pitfalls:
Some learners may read the remainders in the wrong order, starting from the first division instead of the last, which reverses the bits and produces an incorrect binary number. Others may miscalculate powers of 2 when verifying the answer. Remember that the binary equivalent is built by reading remainders from the last division back to the first, and that place values in binary double each step to the left: 1, 2, 4, 8, 16 and so on.

Final Answer:
The binary representation of decimal 101 is 1100101, which corresponds to option D.