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

1. B. 1
2. C. 2
3. D. 3

Answer: C

Explanation

We need to find the distinct remainders of (1ⁿ + 2ⁿ) when divided by 4, where n is a natural number (n ≥ 1).

1. **For 1ⁿ**: Any natural number power of 1 is always 1. So, 1ⁿ mod 4 = 1.

2. **For 2ⁿ**:
* If n = 1, 2¹ = 2. So, 2¹ mod 4 = 2.
* If n = 2, 2² = 4. So, 2² mod 4 = 0.
* If n = 3, 2³ = 8. So, 2³ mod 4 = 0.
* For any n ≥ 2, 2ⁿ will be a multiple of 4 (since 2ⁿ = 4 * 2ⁿ⁻²). So, 2ⁿ mod 4 = 0 for n ≥ 2.

3. **Combining (1ⁿ + 2ⁿ) mod 4**:
* If n = 1: (1¹ + 2¹) mod 4 = (1 + 2) mod 4 = 3 mod 4 = 3.
* If n ≥ 2: (1ⁿ + 2ⁿ) mod 4 = (1 + 0) mod 4 = 1 mod 4 = 1.

Therefore, the distinct remainders are 1 and 3. There are 2 distinct remainders. This question tests basic number theory and modular arithmetic, a common topic in CSAT’s quantitative aptitude section.