Very interesting question!
Let's try to get something that would work for just 5 for now.
Any of these numbers would leave a remainder of 1 when divided by just 5:
6, 11, 16, 21, 26, 31, 36, 41, 46, 51, 56, 61, 66, 71, 76, 81, 86, ...
Notice that they all look like 5*(something) + 1. Basically, they are all multiples of 5 plus 1.
Let's see what would work for just 8:
9, 17, 25, 33, 41, 49, 57, 65, 73, 81, 89, ...
Are there any matches... AHA! It's 81. This is the lowest positive integer greater than 1 that follows the rule for both 5 and 8.