If we let N be the number of cards we are looking for then n= 3i+ 1, n= 4j+ 1, n= 5k+ 1, n= 8l+ 1. (This is equivalent to saying that N is "equal to 1 modulo 3, 4, 5, 8".)

We can, for example, set n= 3i+ 1= 4j+ 1= 5k+ 1= 8l+ 1 so that 3i= 4j= 5k= 8l. From "4j= 8l", we must have = 2l so we can reduce that to 3i= 5k= 8l. Now, what are the smallest possible values of i, k, and l so that those are true? That is, what is the smallest number that k is a multiple of 3, 5, and 8?