Which one of the following is correct?

Concept. In a base-b representation each digit is multiplied by the base raised to its position index, counted from 0 at the rightmost digit, and the value of the number is the sum of these products: (dn-1…d1d0)b = Σ di × bi. The base label is therefore part of the value and not mere decoration — the same digit string denotes different quantities in different bases, and two representations are equal only when their weighted sums come to the same number.

Application. Convert every right-hand side to decimal and compare it with (17)10, whose value is seventeen.

1. (17)16 = 1 × 161 + 7 × 160 = 16 + 7 = 23

2. (17)8 = 1 × 81 + 7 × 80 = 8 + 7 = 15

3. (10111)2 = 16 + 0 + 4 + 2 + 1 = 23

4. (10001)2 = 16 + 0 + 0 + 0 + 1 = 17

Cross-check. Work in the other direction, dividing repeatedly by 2 and recording each remainder:

1. 17 ÷ 2 = 8, remainder 1

2. 8 ÷ 2 = 4, remainder 0

3. 4 ÷ 2 = 2, remainder 0

4. 2 ÷ 2 = 1, remainder 0

5. 1 ÷ 2 = 0, remainder 1

Reading the remainders from the last one to the first gives 10001, which matches the binary form found above.

Result. The equality that holds is (17)10 = (10001)2. Notice also that 23 appears twice in the table, because (17)16 and (10111)2 are two writings of the same quantity.