Mathematician: "It was the August of 1997 when they barged into my house..." - Here is his story:
Students surprised their mathematics teacher with a cake on his birthday. The teacher was delighted when he noticed that every single year of his life is accounted for with a candle. He repeatedly tried to blow out the candles, but he never got all of them. Eventually, he gave up and said: "The youngest of my children, my youngest son, had more luck on his birthday cakes!". One of the students followed up: "How many children do you have and how old are they?"
The teacher decided it is time for a riddle: "The product of their ages is equal to the total number of candles on the cake, while the sum of their ages is equal to the number of candles still left burning on the cake. What can you tell me about my children?"
The students took some time and eventually replied: "We don't know their ages, but we do know that you don't have twins among your children."
The teacher added: "Oh, I forgot to mention that the age of my youngest son is not a cube number."
They replied: "We now know their exact ages!"
Question. What are the ages of the children?