Jason has some baseball cards from the $$1920$$s. If he divides the number of cards he has by $$3$$, the remainder is $1$; if he divides the number of cards he has by $$5$$, the remainder is $3$; if he divides the number of cards he has by $$7$$, the remainder is $5$. Suppose Jason has at least $k$ cards, the sum of digits of $k$ is~\uline{~~~~~~~~~~}~. 
Overseas Competition->Knowledge Point->Number Theory Modules->Factors and Multiples->Common Factors and Common Multiples
The number of cards after adding $$2$$ is divisible by $$3$$, $$5$$, and $$7$$. Since the least common multiple of $$3$$, $$5$$, and $$7$$ is $$3\times5\times7=105$$, Jason has $$105-2=103$$ cards at least. 