smc

Suppose that Chloe is $c$ years old today, so Joey is $c+1$ years old today. After $n$ years, Chloe and Zoe will be $n+c$ and $n+1$ years old, respectively. We are given that $$\frac{n+c}{n+1}=1+\frac{c-1}{n+1}$$ is an integer for $9$ nonnegative integers $n.$ It follows that $c-1$ has $9$ positive divisors. The prime factorization of $c-1$ is either $p^8$ or $p^2{q}^2.$ Since $c-1<100,$ the only possibility is $c-1=2^2\cdot 3^2=36,$ from which $c=37.$ We conclude that Joey is $c+1=38$ years old today. Suppose that Joey's age is a multiple of Zoe's age after $k$ years, in which Joey and Zoe will be $k+38$ and $k+1$ years old, respectively. We are given that $$\frac{k+38}{k+1}=1+\frac{37}{k+1}$$ is an integer for some positive integer $k.$ It follows that $37$ is divisible by $k+1,$ so the only possibility is $k=36.$ We conclude that Joey will be $k+38=74$ years old then, from which the answer is $7+4=\boxed{11}.$