The children are age 4aged $4$ and 1.
4$1$.
$4$ is the only number which can be a product of 2two different combinations of 2two numbers below 6$6$, whilewhere one of these combinations consists of 2two identical numbers.
We
We need 2two combinations because the information ofknowing the product of their ages wasn't enough information.
One
One of these combinations has to consist of 2two identical numbers because the information that there is one child that is older than the other one has to eliminate that combination.
The only product this can refer to is 4$4$ with the combinations 22 and 14$2\times2$ and $1\times4$.