Question

A young boy dropped his deck of baseball cards. Unwilling to count them again he remembered that when he put them in piles of two, there was one card left over. When he put them in stacks of three, there was one card left over. The same thing happened for stacks of 4, 5, 6, but when he put them in stacks of 7, there were no cards left over!

How many cards did the boy have? Is there only one possibility? Can you generalize a solution?

Answer 1

so if he put them in 2 stacks and there was 1 card left over, he has an odd number of cards

when he put them in 3 stacks there was 1 card left over
we can tell that if he took away one of his card and divided
the stack of cards by two or three the number would be the
same

so he divides it by 4 5 and 6 so the number of cards is not divisible by 2,3,4,5,6
but it is divisable by 7
so we must find a number that when you subtract one you can divide it by 2,3,4,5,6
so i just guessed a term from 7^n
so 49 doesn't work because 48 doesn't divide into 5
343 doesn't work because 342 doesn's divide by 6
but 2401 (7^4) works because 2400 divides evenly by 2,3,4,5, and 6

Just a note, this number is not very logical that he would have time to count them all but hey, it's math