If $n$ is a natural number, then what is the number of distinct remainders of $(1^n + 2^n)$ when divided by 4?
If $n$ is a natural number, then what is the number of distinct remainders of $(1^n + 2^n)$ when divided by 4?
0
1
2
3
Solution
$1^n = 1$ always. For $2^n$: $n=1$ gives $2$, $n\ge2$ gives $2^n$ divisible by 4 (remainder 0). So $(1^n+2^n) \bmod 4$: $n=1$ gives $1+2=3$; $n\ge2$ gives $1+0=1$. The distinct remainders are 3 and 1 - that is 2 distinct remainders.