Analyzing Remainders of $1^n + 2^n$ Divided by 4

We are asked to find the number of distinct remainders when the expression $\left(1^n + 2^n\right)$ is divided by 4, where $n$ is a natural number. A natural number $n$ can take values 1, 2, 3, 4, and so on.

Let's evaluate the expression $\left(1^n + 2^n\right)$ for the first few natural numbers and find the remainder when each result is divided by 4.

• For $n = 1$: The expression is $1^1 + 2^1 = 1 + 2 = 3$. When 3 is divided by 4, the remainder is 3.
• For $n = 2$: The expression is $1^2 + 2^2 = 1 + 4 = 5$. When 5 is divided by 4, the remainder is 1 ($5 = 4 \times 1 + 1$).
• For $n = 3$: The expression is $1^3 + 2^3 = 1 + 8 = 9$. When 9 is divided by 4, the remainder is 1 ($9 = 4 \times 2 + 1$).
• For $n = 4$: The expression is $1^4 + 2^4 = 1 + 16 = 17$. When 17 is divided by 4, the remainder is 1 ($17 = 4 \times 4 + 1$).

Let's observe the terms $1^n$ and $2^n$ separately when divided by 4.

• The term $1^n$ is always 1 for any natural number $n$. So, $1^n \pmod{4}$ is always 1.
• The term $2^n$:
  • For $n = 1$, $2^1 = 2$. $2 \pmod{4} = 2$.
  • For $n = 2$, $2^2 = 4$. $4 \pmod{4} = 0$.
  • For $n = 3$, $2^3 = 8$. $8 \pmod{4} = 0$.
  • For $n = 4$, $2^4 = 16$. $16 \pmod{4} = 0$.

For $n \ge 2$, $2^n$ contains a factor of $2^2 = 4$. Thus, for $n \ge 2$, $2^n$ is divisible by 4, which means $2^n \pmod{4} = 0$.

Now let's consider the sum $\left(1^n + 2^n\right) \pmod{4}$ based on the value of $n$:

• When $n = 1$: $\left(1^1 + 2^1\right) \pmod{4} = (1 + 2) \pmod{4} = 3 \pmod{4} = 3$.
• When $n \ge 2$: $\left(1^n + 2^n\right) \pmod{4} = (1^n \pmod{4} + 2^n \pmod{4}) \pmod{4}$. Since $1^n \pmod{4} = 1$ and for $n \ge 2$, $2^n \pmod{4} = 0$, the expression becomes $(1 + 0) \pmod{4} = 1 \pmod{4} = 1$.

So, the possible remainders of $\left(1^n + 2^n\right)$ when divided by 4 are 3 (when $n=1$) and 1 (when $n \ge 2$).

Let's summarize the results in a table:

The distinct remainders we found are 3 and 1. There are exactly two distinct remainders.

Revision Table: Analyzing Number Remainders

Additional Information: Understanding Modular Arithmetic for Remainders

Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after reaching a certain value, called the modulus. The remainder is the result of the modulo operation.

• For example, $7 \pmod{3} = 1$, because $7 = 2 \times 3 + 1$.
• In this problem, the modulus is 4. We are interested in the possible values of $(1^n + 2^n) \pmod{4}$. The possible remainders when dividing by 4 are 0, 1, 2, or 3.
• The behavior of $2^n \pmod{4}$ changes significantly between $n=1$ and $n \ge 2$. This is because $2^1 = 2$, which is not divisible by 4, but $2^n$ for $n \ge 2$ has $2^2=4$ as a factor, making it divisible by 4.
• $1^n$ is always 1, so $1^n \pmod{4}$ is always 1, regardless of the natural number $n$.

By analyzing the expression for different ranges of $n$ (specifically $n=1$ and $n \ge 2$), we could determine all the possible remainder values.