UPSC Prelims 2025 CSAT, question 46

Topic: Number system

Q46

If $n$ is a natural number, then what is the number of distinct remainders of $(1^n+2^n)$ when divided by 4 ?

Solution.

Answer: (c) 2

For $n = 1$ , $1^1 + 2^1 = 3$ , which leaves a remainder of $3$ when divided by $4$ .
For every integer $n \ge 2$ , $2^n = 4 \times 2^{n-2}$ is divisible by $4$ , so $1^n + 2^n = 1 + 2^n \equiv 1 \pmod{4}$ .
Therefore, the only possible remainders are $3$ and $1$ , giving $2$ distinct remainders.