3. What is the remainder when the number $1+2+2^{2}+2^{3}+\ldots+2^{2021}$ is divided by 9?
First method:

The number 1 when divided by 9 gives a remainder of 1.

The number 2 when divided by 9 gives a remainder of 2.

The number $2^{2}$ when divided by 9 gives a remainder of 4.

The number $2^{3}$ when divided by 9 gives a remainder of 8.

The number $2^{4}$ when divided by 9 gives a remainder of 7.

The number $2^{5}$ when divided by 9 gives a remainder of 5.

1 POINT

Since $2^{6}$ gives a remainder of 1 when divided by 9, the remainders of the number $2^{n}$ when divided by 9 repeat periodically with a period of 6.

Therefore, we first calculate the sum of the remainders for the first six addends $1+2+4+8+7+5=27$ and conclude that the sum $1+2+2^{2}+2^{3}+2^{4}+2^{5}$ is divisible by 9.

2 POINTS

Since $2^{6+k}=2^{6} \cdot 2^{k}$, $2^{6+k}$ for any non-negative integer $k$ gives the same remainder when divided by $k$ as $2^{k}$, i.e., since the remainders of the number $2^{n}$ when divided by 9 repeat periodically with a period of 6, we conclude that the sum $2^{6 k}+2^{6 k+1}+2^{6 k+2}+2^{6 k+3}+2^{6 k+4}+2^{6 k+5}$ is divisible by 9 for any non-negative integer $k$. 3 POINTS

The sum $1+2+2^{2}+2^{3}+\ldots+2^{2021}$ has 2022 addends.

1 POINT

Since 2022 is divisible by 6,

1 POINT the addends in the original sum can be grouped into sets of 6 addends (there are 337 such sets because $2022: 6=337$) whose sum is divisible by 9

1 POINT and thus we conclude that the original sum is divisible by 9, i.e., the remainder when the number $1+2+2^{2}+2^{3}+\ldots+2^{2021}$ is divided by 9 is equal to 0.

1 POINT

Note: Instead of writing generally that each sum $2^{6 k}+2^{6 k+1}+2^{6 k+2}+2^{6 k+3}+2^{6 k+4}+2^{6 k+5}$ for any non-negative integer $k$ is divisible by 9, the student can write that, because the remainders of the number $2^{n}$ when divided by 9 repeat periodically with a period of 6, each of the sums $1+2+2^{2}+2^{3}+2^{4}+2^{5}$, $2^{6}+2^{7}+2^{8}+2^{9}+2^{10}+2^{11}$, $2^{12}+2^{13}+2^{14}+2^{15}+2^{16}+2^{17}$, etc., is divisible by 9.

TOTAL 10 POINTS