Q. If n is a natural number, then what is the number of distinct remainders of (1ⁿ + 2ⁿ) when divided by 4?
(a) 0
(b) 1
(c) 2
(d) 3
Correct Answer : (c) 2
UPSC Prelims 2025 CSAT
Explanation :
1. Observe that 1ⁿ is always 1, whatever n is.
2. So we need only look at 2ⁿ mod 4, then add 1 and reduce mod 4.
– n = 1: 2¹ = 2 ⇒ 2 mod 4 = 2 ⇒ 1 + 2 = 3 mod 4
– n = 2: 2² = 4 ⇒ 4 mod 4 = 0 ⇒ 1 + 0 = 1 mod 4
– n = 3: 2³ = 8 ⇒ 8 mod 4 = 0 ⇒ 1 + 0 = 1 mod 4
– For any n ≥ 2, 2ⁿ is a multiple of 4, so 2ⁿ mod 4 = 0, giving 1 + 0 = 1 mod 4.
3. Hence the only remainders that occur are 3 (when n = 1) and 1 (for all n ≥ 2).
Number of distinct remainders = 2.
