Brainstorming with Hats

Logic Level 2

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, \(N,\) written on it, and that the other two hats each have a divisor of \(N\) written on them. Each person can only see the two numbers on the others’ hats.

Jane: "I don't know my number."
Kate: "I don't know my number."
Lily: "But I know my number!"
Kate: "Now I know my number, too!"
Jane: "I still don't know my number."

What is the sum of the numbers written on Kate and Lily’s hats?

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