其它
Leo prepares more than $400$ cupcakes for a party. Now he can divided all of them equally into $5$ piles. After his pet cat eats one of the cupcakes, he finds that now he can divide the remaining cupcakes equally into $6$ piles. Then, the naughty pet cat eat another one, and Leo divides the remaining cupcakes into $7$ piles. How many cupcakes did Leo make at least at the beginning? Find the sum of the three digits. 
Overseas Competition->Knowledge Point->Number Theory Modules->Remainder Problems->Chinese Remainder Theorem
The number is a multiple of $5$. When it is divided by $6$, it will have a remainder of $1$; when it is divided by $7$, it will have a remainder of $2$.  Thus, if we add another $5$ to the number, it can be multiple of all the three numbers, which is $5\times6\times7=210$ at least. But the number is more than $400$, so we can get $210\times2-5=415.$ 