Solution:

At first glance this puzzle seems unsolvable. But it has something weird, and this is the fact of “the last digit of the product of their ages” being a clue. How can something so vague be a clue? Unless it’s not that vague. Let’s explore it a little bit.

We make a table with all possible ages for each daughter and calculate the last digit of their product - i.e. (a*b).mod(10)

We can get rid of half the table (since 6x4 = 4x6) and since the problem says they are not twins, we can also get rid of the main diagonal, the one with the square numbers.

Now let's evaluate each digit on the table, and see what we get.

Let’s start with digit 1: it appears once (on 7x3) and the sum of the factors is 10.

Digit 2 on the other hand, appears 6 times (on 2x1, 4x3, 6x2, 7x6, 8x4 and 9x8) and the sum of factors is 3, 7, 8, 13, 12 and 17

Here is the final table

So, in order for the final digit to be a clue, it must tell us something unique. Looking at this table, we realize only 2 digits fulfill this, the digits 1 and 9. So, if the last digit of the product of their ages is 1, that means they are 7 and 3 years old. And if the final digit is 9, that means they are 9 and 1 years old.

Now we have two possible answers, how to proceed? Well, the problem is not to guess their daughters age, but the sum of it. And since in both cases the answer is 10, that’s your answer.