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

To find the number of distinct remainders of (1^n + 2^n) when divided by 4, we analyze the expression for different values of n, where n is a natural number (n = 1, 2, 3, ...).

1. Analyze 1^n: For any natural number n, 1^n is always 1.

2. Analyze 2^n:

   • If n = 1: 2^1 = 2.
   • If n = 2: 2^2 = 4. When 4 is divided by 4, the remainder is 0.
   • If n = 3: 2^3 = 8. When 8 is divided by 4, the remainder is 0.
   • For any n >= 2, 2^n will be a multiple of 4 (since 2^n = 2^2 * 2^(n-2) = 4 * 2^(n-2)). Therefore, for n >= 2, 2^n leaves a remainder of 0 when divided by 4.

Now, let's combine these for (1^n + 2^n) mod 4:

• Case 1: n = 1 (1^1 + 2^1) = 1 + 2 = 3. When 3 is divided by 4, the remainder is 3.

• Case 2: n >= 2 (1^n + 2^n) = 1 + (a multiple of 4) When (1 + 2^n) is divided by 4, the remainder is (1 + 0) = 1.

So, the possible distinct remainders are 3 (when n=1) and 1 (when n>=2). There are 2 distinct remainders.

The final answer is C.