17. The number 2021 has a remainder of 5 when divided by 6 , by 7 , by 8 , or by 9 . How many positive integers are there, smaller than 2021 , that have this property?
A 4
B 3
C 2
D 1
E none
SolUtion
A

For $N$ to have remainder 5 when divided by $6,7,8$ or 9 , we need $N-5$ to be divisible by 6,7 , 8 and 9 . The $\mathrm{LCM}$ of $6,7,8,9$ is $2^{3} \times 3^{2} \times 7=504$. Hence $N-5$ is a multiple of 504 , so $N-5=0,504,1008,1512,2016 \ldots$ Then $N=5,509,1013,1517,2021 \ldots$ Of these, only the first four are smaller than 2021 and have the stated property.