To solve the division of the binary numbers [tex]\((110111)_2 \div (101)_2\)[/tex], we can follow these steps:
1. Convert the binary numbers to decimal:
- For [tex]\( (110111)_2 \)[/tex]:
- Starting from the right, we can label each bit with the appropriate power of 2:
[tex]\[
1 \cdot 2^5 + 1 \cdot 2^4 + 0 \cdot 2^3 + 1 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0
\][/tex]
- This simplifies to:
[tex]\[
32 + 16 + 0 + 4 + 2 + 1 = 55
\][/tex]
- Thus, [tex]\( (110111)_2 = 55 \)[/tex] in decimal.
- For [tex]\( (101)_2 \)[/tex]:
- Starting from the right, we can label each bit similarly:
[tex]\[
1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0
\][/tex]
- This simplifies to:
[tex]\[
4 + 0 + 1 = 5
\][/tex]
- Thus, [tex]\( (101)_2 = 5 \)[/tex] in decimal.
2. Perform the division in decimal:
- We now have to divide 55 by 5:
[tex]\[
55 \div 5 = 11
\][/tex]
3. Interpret the result:
- The result of the division in decimal is 11.
4. Convert the result back to binary:
- To convert 11 from decimal to binary:
- Find the highest power of 2 less than or equal to 11, which is [tex]\(2^3 = 8\)[/tex].
- Subtract 8 from 11, leaving 3. The next highest power of 2 is [tex]\(2^1 = 2\)[/tex].
- Subtract 2 from 3, leaving 1. The next highest power of 2 is [tex]\(2^0 = 1\)[/tex].
- The binary representation of 11 is [tex]\(1011\)[/tex]:
[tex]\[
11 \div 2 = 5 \text{ remainder } 1 \quad \text{(least significant bit)}
\][/tex]
[tex]\[
5 \div 2 = 2 \text{ remainder } 1
\][/tex]
[tex]\[
2 \div 2 = 1 \text{ remainder } 0
\][/tex]
[tex]\[
1 \div 2 = 0 \text{ remainder } 1 \quad \text{(most significant bit)}
\][/tex]
Therefore, the final answer in binary is:
[tex]\[
(110111)_2 \div(101)_2 = (1011)_2
\][/tex]
This completes the detailed step-by-step solution for the binary division problem. The entire process verifies that:
[tex]\[ \boxed{(110111)_2 \div (101)_2 = (1011)_2} \][/tex]
