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 ✓ UPSC's answer
- (d) 3
Why the answer is (c)
• 1ⁿ is always congruent to 1 modulo 4 for every natural number n.
• 2¹ is congruent to 2 modulo 4, while 2ⁿ is congruent to 0 modulo 4 for every n ≥ 2.
• For n = 1, (1¹ + 2¹) = 3, giving remainder 3 on division by 4.
• For n ≥ 2, (1ⁿ + 2ⁿ) ≡ 1 + 0 = 1, giving remainder 1 on division by 4.
• The distinct remainders are 1 and 3, so the number of distinct remainders is 2, which is option (c).
Why the other options are wrong
- (a) 0
- The expression produces remainders 1 and 3, so the count cannot be 0.
- (b) 1
- The expression gives two different remainders, 3 for n = 1 and 1 for n ≥ 2, so the count is not 1.
- (d) 3
- Only the remainders 1 and 3 occur, so the count is not 3.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2025, held on 25 May 2025. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.