If a natural number can be written as the sum of both two and three consecutive natural numbers, then we can call it a Think Number. What is the largest Think Number no larger than $2022$? 
Overseas Competition->Knowledge Point->Number Theory Modules
The number can be written as $$n+(n+1)=2n+1(n\geqslant 1)$$ and $$x+(x+1)+(x+2)=3x+3$$.  It must be a multiple of $3$ but leaves a remainder of $1$ when divided by $2$.  $2022$ can be divisible by both $2$ and $3$, so it is $2022-3=2019$. 