If (L)M represents a number L in the base-M number system, then which of the following equalities are true? Choose the correct answer from the options given below.

1.125)10 = (1000000000.001)2

2.125)10 = (1000.1)8

3.125)10 = (200.2)16

4.125)10 = (2000.02)4

A1 and 2 only

B1 and 3 only

C2, 3 and 4 only

D1, 2 and 3 only ✓ Correct

Correct answer: (D) 1, 2 and 3 only

Explanation

Converting 512.125 to binary gives 1000000000.001, since 512 is 2 to the ninth power and 0.125 is one-eighth.

Statement 1 holds because the whole part is a single 1 followed by nine zeros and the fraction is 0.001.

Statement 2 holds because 512 in octal is 1000 and 0.125 equals one-eighth, written 0.1.

Statement 3 holds because 512 in hexadecimal is 200 and 0.125 equals two-sixteenths, written 0.2.

Statement 4 fails because 512 in base four is 20000, not 2000, so that equality is false.

A fraction converts cleanly when it is a simple power of the target base's building fraction.

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