2018 AMC 10B Problem 19

Let Chloe be $n$ today; Zoe is $1.$ In $t$ years their age ratio is $$\dfrac{n+t}{1+t}=1+\dfrac{n-1}{1+t},$$ which is an integer exactly when $1+t$ divides $n-1.$ Thus $n-1$ has exactly $9$ positive divisors. A number with $9$ divisors has the form $p^8$ or $p^2q^2.$ The only two-digit possibility is $2^2\cdot3^2=36,$ so Chloe is $37$ and Joey is $38.$ Joey’s age $38+t$ is a multiple of Zoe’s age $1+t$ exactly when $1+t$ divides $37.$ The next time is $t=36,$ when Joey is $74.$ Its digit sum is $7+4=11.$ Thus, E is the correct answer.