10. How many of the numbers $6,7,8,9,10$ are factors of the sum $2^{2024}+2^{2023}+2^{2022}$ ?
A 1
B 2
C 3
D 4
E 5
Solution
B

For convenience, we put $S=2^{2024}+2^{2023}+2^{2022}$.
We have
$$
S=2^{2024}+2^{2023}+2^{2022}=2^{2022}\left(2^{2}+2^{1}+1\right)=2^{2022}(4+2+1)=2^{2022} \times 7 .
$$

It follows that 7 is a factor of $S$. Also, since $8=2^{3}, 8$ is a factor of $2^{2022}$ and hence it is a factor of $S$. On the other hand, 3 and 5 are neither factors of $2^{2022}$ nor factors of 7 . So they are not factors of $S$. It follows that 6 and 9 which are multiples of 3 are not factors of $S$. Similarly, 10 is a multiple of 5 and hence it is not a factor of $S$.
We therefore see that just two of the numbers $6,7,8,9$ and 10 are factors of $S$, namely 7 and 8 .