Question

5
Two mathematicians, Ali and Ismail were chatting. Ismail says he has three children who all have the same birthday (but who weren't necessarily born in the
same year). Ali asks their ages. Ismail replies, "The product of the ages of my children is 72". Ali points out that this information is not enough to determine
their ages. Ismail responds with another clue, he tells Ali the sum of the ages of his children. But Ali again points out that there is not enough information.
Finally Ismail says, "My youngest child is named Ganesh." At last, Ali correctly determines the ages of Ismail's children. What is the age of his youngest child?
A Mark as Error​

Answer 1

Answer:

The youngest child is 2 years old.

Step-by-step explanation:

Note that $72 = 2^3\cdot 3^2$, so we can list all possible triple of natural numbers whose product is 72 arranging  these prime factors.

They are: $\{1,1,72\}, ~\{1,2,36\}, ~\{1,3,24\}, ~\{1,4,18\}, ~\{1,6,12\}, ~\{1,8,9\}\\~\{2,2,18\}, ~\{2,3,12\}, ~\{2,4,9\}, ~\{2,6,6\}, ~\{3,3,8\}, ~\{3,4,6\}.$

From all the possibilities, only two of them have the same sum: $\{2,6,6\}$ and $\{3,3,8\}$. So their ages must be one of these two possibilities, otherwise Ali would have figured out their ages when Ismail told him the sum of the ages.

When Ismail finally says the name of his youngest child, in particular, he says there is a youngest child. From the two possibilities, only the first one has a lower age, that is, their ages are 2, 6 and 6. So his youngest child is 2 years old.