If (L)m represents a number L in base-M number system, then which of the following equalities are true?

A) (127.125)10 = (1111111.001)2
B) (127.125)10 = (1333.02)4
C) (127.125)10 = (177.1)8
D) (127.125)10 = (7F.2)16Choose the most appropriate answer from the options given below.

The correct option is A, B, C and D.

To verify the equalities, we convert the decimal number ${\left(127.125\right)}_{10}$ into base 2, base 4, base 8, and base 16. We divide this process into two parts: the integer part (127) and the fractional part (0.125).

Part 1: Converting the Integer Part (127)₁₀

1. Conversion to Binary (Base 2):
We repeatedly divide 127 by 2 and note the remainders:
127 ÷ 2 = 63, remainder 1
63 ÷ 2 = 31, remainder 1
31 ÷ 2 = 15, remainder 1
15 ÷ 2 = 7, remainder 1
7 ÷ 2 = 3, remainder 1
3 ÷ 2 = 1, remainder 1
1 ÷ 2 = 0, remainder 1
Reading the remainders from bottom to top, we get:
${\left(127\right)}_{10}={\left(1111111\right)}_2$

2. Conversion to Base 4:
We repeatedly divide 127 by 4:
127 ÷ 4 = 31, remainder 3
31 ÷ 4 = 7, remainder 3
7 ÷ 4 = 1, remainder 3
1 ÷ 4 = 0, remainder 1
Reading from bottom to top, we get:
${\left(127\right)}_{10}={\left(1333\right)}_4$

3. Conversion to Octal (Base 8):
We repeatedly divide 127 by 8:
127 ÷ 8 = 15, remainder 7
15 ÷ 8 = 1, remainder 7
1 ÷ 8 = 0, remainder 1
Reading from bottom to top, we get:
${\left(127\right)}_{10}={\left(177\right)}_8$

4. Conversion to Hexadecimal (Base 16):
We repeatedly divide 127 by 16:
127 ÷ 16 = 7, remainder 15 (which is F in hexadecimal)
7 ÷ 16 = 0, remainder 7
Reading from bottom to top, we get:
${\left(127\right)}_{10}={\left(\mathrm{7F}\right)}_{16}$

Part 2: Converting the Fractional Part (0.125)₁₀

We repeatedly multiply the fractional part by the target base and record the integer parts of the results.

1. Conversion to Binary (Base 2):
0.125 × 2 = 0.25 (integer part: 0)
0.25 × 2 = 0.5 (integer part: 0)
0.5 × 2 = 1.0 (integer part: 1)
Reading the integer parts from top to bottom, we get:
${\left(0.125\right)}_{10}={\left(0.001\right)}_2$

2. Conversion to Base 4:
0.125 × 4 = 0.5 (integer part: 0)
0.5 × 4 = 2.0 (integer part: 2)
Reading the integer parts from top to bottom, we get:
${\left(0.125\right)}_{10}={\left(0.02\right)}_4$

3. Conversion to Octal (Base 8):
0.125 × 8 = 1.0 (integer part: 1)
Reading the integer parts from top to bottom, we get:
${\left(0.125\right)}_{10}={\left(0.1\right)}_8$

4. Conversion to Hexadecimal (Base 16):
0.125 × 16 = 2.0 (integer part: 2)
Reading the integer parts from top to bottom, we get:
${\left(0.125\right)}_{10}={\left(0.2\right)}_{16}$

Conclusion:
Combining the integer and fractional parts for each base, we get:
A) ${\left(127.125\right)}_{10}={\left(1111111.001\right)}_2$ (True)
B) ${\left(127.125\right)}_{10}={\left(1333.02\right)}_4$ (True)
C) ${\left(127.125\right)}_{10}={\left(177.1\right)}_8$ (True)
D) ${\left(127.125\right)}_{10}={\left(\mathrm{7F.2}\right)}_{16}$ (True)
Therefore, all four equalities A, B, C, and D are true.