The logic offered here follows the deduction process.
The product must be some number $1 \leq p \leq 25$, where $p = a_1 a_2$ with $1 \leq a_1 \leq a_2 \leq 5$. The $a_i$ are the ages.
If $p$ were prime, then we'd have $a_1 = 1$ and $a_2 = p$ so $p$ may not be prime, because the product information is insufficient. We thus know $p$ may not be $2,3,5,7,11,13,17,19,23$. We may also exclude product values that, when factorized, necessarily include a factor greater than 5. This way, we also exclude $14,18,21,22,24$.
Because the product value $p$ is at first insuficcient, it must admit more than one factorization in factors no greater than 5. This excludes $6, 8, 9, 10, 12, 15, 16, 20, 25$. The only possibility is thus $4 = 2\times 2 = 4\times 1$. Since there must be an elder, they may not be equal factors, and the ages are $1$ and $4$.
EDIT: Welp, a little too late.