17. A list of integers has a sum of 2018 , a product of 2018 , and includes the number 2018 in the list. Which of the following could be the number of integers in the list?
A 2016
B 2017
C 2018
D 2019
E 2020
17. B Let the integers be $\left\{2018, a_{1}, a_{2}, \ldots, a_{n}\right\}$. Their product is $2018 \times a_{1} \times a_{2} \times \ldots \times a_{n}=2018$ so we get $a_{1} \times a_{2} \times \ldots \times a_{n}=1$, meaning each $a_{i}$ is either 1 or -1 (with an even number of occurrences of -1 ).
Their sum is $2018+a_{1}+a_{2}+\ldots+a_{n}=2018$ so $a_{1}+a_{2}+\ldots+a_{n}=0$, meaning an equal number of occurrences of 1 and -1 . Hence the list includes 2018 once, an even number of $1 \mathrm{~s}$ and the same even number of $-1 \mathrm{~s}$. Therefore the number of integers must be one more than a multiple of 4 , so from the list of options only 2017 is possible.