Which of the following is equivalent to `(125)_(10)` ?

Options

A. `(1331)4`
B. `(175)8`
C. `(1111101)2`
D. All of the above

Answer

To determine which of the given options is equivalent to \( (125)_{10} \), we will convert each option from its respective base to decimal (base 10) and check if it equals 125.

### Step 1: Convert Option 1 from Base 4
The first option is \( (1331)_4 \).

To convert from base 4 to decimal:
\[ (1331)_4 = 1 \times 4^3 + 3 \times 4^2 + 3 \times 4^1 + 1 \times 4^0 \]

Calculating each term:
- \( 1 \times 4^3 = 1 \times 64 = 64 \)
- \( 3 \times 4^2 = 3 \times 16 = 48 \)
- \( 3 \times 4^1 = 3 \times 4 = 12 \)
- \( 1 \times 4^0 = 1 \times 1 = 1 \)

Now, add these values together:
\[ 64 + 48 + 12 + 1 = 125 \]

So, \( (1331)_4 = (125)_{10} \).

### Step 2: Convert Option 2 from Base 8
The second option is \( (175)_8 \).

To convert from base 8 to decimal:
\[ (175)_8 = 1 \times 8^2 + 7 \times 8^1 + 5 \times 8^0 \]

Calculating each term:
- \( 1 \times 8^2 = 1 \times 64 = 64 \)
- \( 7 \times 8^1 = 7 \times 8 = 56 \)
- \( 5 \times 8^0 = 5 \times 1 = 5 \)

Now, add these values together:
\[ 64 + 56 + 5 = 125 \]

So, \( (175)_8 = (125)_{10} \).

### Step 3: Convert Option 3 from Base 2
The third option is \( (111101)_2 \).

To convert from base 2 to decimal:
\[ (111101)_2 = 1 \times 2^5 + 1 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 \]

Calculating each term:
- \( 1 \times 2^5 = 1 \times 32 = 32 \)
- \( 1 \times 2^4 = 1 \times 16 = 16 \)
- \( 1 \times 2^3 = 1 \times 8 = 8 \)
- \( 1 \times 2^2 = 1 \times 4 = 4 \)
- \( 0 \times 2^1 = 0 \times 2 = 0 \)
- \( 1 \times 2^0 = 1 \times 1 = 1 \)

Now, add these values together:
\[ 32 + 16 + 8 + 4 + 0 + 1 = 61 \]

So, \( (111101)_2 \neq (125)_{10} \).

### Conclusion
The options that are equivalent to \( (125)_{10} \) are:
- Option 1: \( (1331)_4 \)
- Option 2: \( (175)_8 \)