Lucy Becomes Crazier! (FT Edition)

Number Theory Level 5

A surveyor named Mystogan comes to make data all the ages of people living in a housing complex. He entered to the complex. He noticed that the house had been arranged for only one row. He saw number \(25\) on one of the house. He knew that it was the largest number in that complex since he had lived there once. He started to collect all the ages of the residents from the first house he saw.

Then, he visited to Natsu and Lucy's house. Both of them had three children, named Happy, Nappy and Zappy. Lucy and Natsu was outside and saw Mystogan.

Here is the dialog:

Mystogan: Hello, madam. Can I have a favor?

Lucy: Yes. What's that?

Mystogan: How old are your three children

Natsu: Oh.... I forgot about it also.

Lucy: Natsu, you always go battling the evils and now you forget even the ages of your children. I hope you won't forget your children's name. I won't tell the ages directly, but I'll give you some clues

Mystogan and Natsu: What's that?

Lucy: The product of three children ages are \(144\).

Mystogan: I can't get any. Any clues?

Lucy: The sum of my three children is equal to when you had tied your shoes in front of a house before you had arrived here. You've never passed here before. I saw you since Natsu and I were chattering about our guilds.

Mystogan thought a while. However he couldn't answer either.

Mystogan: I can't answer it either. Any clues?

Natsu: Oh ya. I've been reminded the age of my children right now.. Ehh.. I don't know

Lucy: HEY!! HEY!!! YOU IDIOT (to Natsu). This is the last clue. You will play a number game. Choose two out of three ages of my children, use ONLY those numbers you choose and operate only one operator (either add, subtract, times or divide). It's result must be also the ages of my children too (either Zappy, Nappy or Happy).

Mystogan: Owh!! I've got it!!!

Question: What is the number of the house where Mystogan tied his shoes before surveying Natsu and Lucy's children? How old is the oldest children? State in sum of both solution


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