Convert the decimal number 61 (base 10) to its binary (base 2) representation. Show the place-value or division-by-2 reasoning.

Introduction / Context:
Converting between decimal and binary is a routine skill in digital design and programming. It reinforces understanding of positional notation and is useful for interpreting register values, masks, and addresses. Here we convert 61 from base 10 to base 2.

Given Data / Assumptions:

• Value: 61 decimal.
• Target: binary representation without leading zeros.
• Allowed method: place-value expansion or repeated division by 2.

Concept / Approach:
Use place-value expansion: represent 61 as a sum of powers of two. Alternatively, perform repeated division by 2 and read remainders bottom-up. Both methods should produce the same bit pattern. The highest power of two not exceeding 61 is 32 (2^5).

Step-by-Step Solution:
List powers: 2^5=32, 2^4=16, 2^3=8, 2^2=4, 2^1=2, 2^0=1.Select terms that sum to 61: 32 + 16 + 8 + 4 + 1 = 61.Mark bits: for 2^5,2^4,2^3,2^2,2^1,2^0 → 1,1,1,1,0,1.Write bits from 2^5 down to 2^0: 111101.

Verification / Alternative check:
Division method: 61/2=30 r1, 30/2=15 r0, 15/2=7 r1, 7/2=3 r1, 3/2=1 r1, 1/2=0 r1. Reading remainders upward yields 111101, which matches the place-value result.

Why Other Options Are Wrong:
111100 equals 60; 110011 equals 51; 001101 equals 13; “None” is wrong because 111101 is correct.

Common Pitfalls:
Dropping a remainder in the division method; writing bits in reverse order; confusing the highest power needed and leaving a gap in the expansion.

Final Answer:
111101