Jane, Kate, and Lily are perfect logicians and always tell the truth.
They are each wearing a hat marked with a distinct positive integer. They know that one of the hats has a single-digit integer, written on it, and that the other two hats each have a divisor of written on them. Each person can only see the two numbers on the others’ hats.
What is the sum of the numbers written on Kate and Lily’s hats?