Which one of the following represents the binary equivalent of the decimal…

Concept — A binary numeral is written in base 2. Counting positions from 0 at the right-hand end, a digit d standing at position i contributes d × 2i to the total, and the only digits available are 0 and 1, so each power of two is either present (1) or absent (0).

Method — To convert a decimal number into base 2, divide it by 2 repeatedly and record the remainder (0 or 1) at every step; the remainders, read from the last one obtained back to the first, spell the binary string. The same result follows from writing the number as a sum of distinct powers of 2.

Application to the number 25

1. 25 ÷ 2 = 12, remainder 1

2. 12 ÷ 2 = 6, remainder 0

3. 6 ÷ 2 = 3, remainder 0

4. 3 ÷ 2 = 1, remainder 1

5. 1 ÷ 2 = 0, remainder 1

Reading the remainders from the last one obtained back to the first gives 11001. The power-sum route agrees: 25 = 16 + 8 + 1 = 24 + 23 + 20, so positions 4, 3 and 0 carry 1 while positions 2 and 1 carry 0, which again spells 11001.

Cross-check — expanding that string by place value gives 1×24 + 1×23 + 0×22 + 0×21 + 1×20 = 16 + 8 + 0 + 0 + 1 = 25, which is exactly the decimal number asked for.

Contrast — the four five-digit strings offered stand for different decimal values:

So the string worth 25 is 11001, the binary equivalent of decimal 25.