Joey and Chloe and their daughter Zoe all have the same birthday. Joey is $1$ year older than Chloe, and Zoe is exactly $1$ year old today. Today is the first of the $8$ birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the digits of Joey's age the next time his age is a multiple of Zoe's age?