Question

The number \( 100101_2 \) is equivalent to octal

• A. 54
• B. 45
• C. 37
• D. 25

Hint:

Since \( 2^3 = 8 \), every single octal digit precisely represents a unique 3-bit binary sequence. Memorizing the 3-bit sequences from 000 to 111 speeds up these conversions significantly.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given a number in the binary number system (base 2) and need to convert it into its equivalent value in the octal number system (base 8).


Step 2: Key Formula or Approach:
To convert a binary number to an octal number, group the binary digits into sets of three, starting from the rightmost digit (the least significant bit).
Then, convert each 3-bit group into its equivalent decimal/octal digit.


Step 3: Detailed Explanation:
The given binary number is \( 100101_2 \).
Let's group the digits into blocks of three starting from the right:
\[ \underbrace{100}_{\text{Group 2}} \ \underbrace{101}_{\text{Group 1}} \] Now, evaluate the decimal value of each group.
For the first group on the right (\( 101 \)):
\[ 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 4 + 0 + 1 = 5 \] For the second group on the left (\( 100 \)):
\[ 1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 = 4 + 0 + 0 = 4 \] Combining these evaluated digits from left to right, we get \( 45 \).
Therefore, \( (100101)_2 = (45)_8 \).


Step 4: Final Answer:
The octal equivalent is 45.

Answer 2

This question asks for the octal equivalent of the binary number \( 100101_2 \). Instead of grouping the bits directly into sets of three, this method takes a two-hop route: first convert the binary number to decimal, then convert that decimal value to octal.

Expand the binary number using powers of 2, starting from the rightmost bit at \( 2^0 \):

\[ 100101_2 = (1 \times 2^5) + (0 \times 2^4) + (0 \times 2^3) + (1 \times 2^2) + (0 \times 2^1) + (1 \times 2^0) \]\[ = 32 + 0 + 0 + 4 + 0 + 1 = 37 \]

So the decimal equivalent of \( 100101_2 \) is \( 37 \).

Now convert \( 37 \) to octal using successive division by 8, recording the remainders:

\[ 37 \div 8 = 4 \text{ remainder } 5 \]\[ 4 \div 8 = 0 \text{ remainder } 4 \]

Reading the remainders from bottom to top gives \( 45_8 \).

Taking this decimal detour lands on exactly the same result as grouping the bits directly, which confirms the conversion is correct.

Therefore, the correct answer is 45.