How old are the kids?

Two mathematicians met many, many years.
- How old are your children?
asks one. What the other answer:
- Product of the 3 years is 36, and the sum of the years is the number on the opposite house.
- Well, unfortunately, I still do not know the answer.
- All right, I also tell you that the greatest loves spinach!
- I already know!
Cried the asker.
How old are the children?

how old are the children

Solution:

If you look at how the 36 can be divided into 3 numbers that the product is 36 of these numbers and calculate the sum of these numbers, it is shown that the sum in two cases will be 13. Therefore, 13 may be the only the number on the opposite house, because all the other sum are different. The sum is 13 if the years are 1, 6, 6 or 2, 2, 9. The last one has the largest number, so the children are 2 and 2 and 9 years old.

Note:

The problem is interesting, because to find how year old the children, you should be understand what is in the background of the text of the problem.
You can surprise to hear about the number on the opposite house, that you can’t see, rather than the greatest child likes the spinach. Many people feel that there is something trick and not a real math problem.

You have to find all 3 numbers that the product should be 36:

36 = 1 x 1 x 36 (38)
36 = 1 x 2 x 18 (21)
36 = 1 x 3 x 12 (16)
36 = 1 x 4 x 9 (14)
36 = 1 x 6 x 6 (13)
36 = 2 x 2 x 9 (13)
36 = 2 x 3 x 6 (11)
36 = 3 x 3 x 4 (10)

The mathematician see the number on the opposite house, so he know the sum of numbers. If the opposite house number would be 11, for example, the solution of the problem could be only 2, 3 and 6, but in that case mathematician would not say: “Well, unfortunately, I still do not know the answer”. Therefore, the sum shouldn’t be unique number.
That means we can say the number of the opposite house although it wasn’t question.
The additional condition is obviously not the information that the child likes the spinach, but the biggest is exist among them. And that’s enough to be able to choose the correct one of the two potential possibilities.