### Instruction: Emily ordered her playing cards by suit in the order $$A,2,3,4,5,6,7,8,9,10,J,Q,K,A,2,3,\cdots.$$What is the $42$nd card?### Response: I need to find a pattern or a rule that relates the position of a card to its suit and value.
I notice that there are $13$ cards of each suit, and they cycle through the suits in alphabetical order: clubs, diamonds, hearts, and spades.
So every $13$ cards, I start a new suit, and every $52$ cards, I start a new cycle of suits.
I can use division and remainder to find out which suit and which cycle I'm in for the $42$nd card.
If I divide $42$ by $13$, I get $3$ with a remainder of $3$.
That means I'm in the third suit, which is hearts, and I'm in the third position of that suit, which is $3$.
So the $42$nd card is the $3$ of hearts.