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

The correct option is 2.

Let us determine the distinct remainders when the expression $1^n+2^n$ is divided by 4, where $n$ is a natural number ($n\in \mathbb{N}$, so $n=1,2,3,\dots$).

First, observe that for any natural number $n$:

$1^n=1$

So the given expression simplifies to:

$E\left(n\right)=1+2^n$

Now, let us evaluate $E\left(n\right)mod4$ for different values of $n$:

Case 1: When $n=1$

$E\left(1\right)=1+2^1=1+2=3$

When 3 is divided by 4, the remainder is 3.

Case 2: When $n\ge 2$

For any natural number $n\ge 2$, $2^n$ can be written as $4\times 2^{n-2}$, which is a multiple of 4.

Therefore, $2^n\equiv 0(mod4)$ for all $n\ge 2$.

Substituting this back into the expression:

$E\left(n\right)=1+2^n\equiv 1+0\equiv 1(mod4)$

When divided by 4, the remainder for all $n\ge 2$ is 1.

Conclusion:

The possible remainders when $(1^n+2^n)$ is divided by 4 are 3 (for $n=1$) and 1 (for $n\ge 2$).

Thus, the set of distinct remainders is {1, 3}, which means there are 2 distinct remainders in total.