Which of the following statements is/are correct? I. (1071)10 = (2057)8 II.…

Concept — positional value and base conversion

A numeral written in base b is a positional sum: read from the right, its digits carry the weights b0, b1, b2, and so on. Two numerals written in different bases denote the same number exactly when these expansions evaluate to the same value.

In the other direction, converting a base-10 number into base b is repeated division by b — divide, record the remainder, keep dividing the quotient, and read the remainders from the last division upward.

Application — statement I: decimal to octal

1. 1071 ÷ 8 = 133, remainder 7

2. 133 ÷ 8 = 16, remainder 5

3. 16 ÷ 8 = 2, remainder 0

4. 2 ÷ 8 = 0, remainder 2

Reading the remainders from the last division upward gives the octal numeral 2057, so the equality asserted in statement I is a true equality.

Application — statement II: hexadecimal to decimal

In hexadecimal A = 10 and F = 15, and the three digit positions carry the weights shown below.

The expansion totals 256 + 160 + 15 = 431, and 431 is not equal to 413, so the equality asserted in statement II is not a true equality.

Cross-check

• Expanding the octal numeral 2057 the other way gives 2 × 83 + 0 × 82 + 5 × 81 + 7 × 80 = 1024 + 0 + 40 + 7 = 1071, which reproduces the decimal value used in statement I.

• Conversely, the decimal number 413 is written 19D in hexadecimal, not 1AF — a second route to the same conclusion about statement II.

Only statement I is a true equality, so the response that matches is "Only I".