Xuomi wants to convert his two storey house that has 5 rooms each in two floors into a hotel. Xuomi decides to number his first floor rooms with 5 consecutive numbers in the 100's, and his second floor numbers would be 5 consecutive numbers in the 200's. Goes without saying that the 5 room numbers in the 2nd floor are 100 more than the corresponding room numbers from the first floor. Being a mathematician, Xuomi is extremely fascinated by primes and also unconcerned about other people's preference for simple sequences. So, he decides to number his rooms in such a way as to have as many prime numbers as possible. His friend, Pytha told him blithely that the maximum he could manage was 5 numbers. Xuomi generated the maximum amount of contempt possible with his eyes and told his friend that not even his father (Pytha's father, not Xuomi's) Goras could accommodate 5 primes and 4 primes was the limit of anyone's ambition. So, they decided to go for 4. Pytha suggested that they might want to go for numbers such that average number of all the room numbers put together would also be prime. Xuomi checked his math and told him this was possible. Xuomi also wanted the first room, or the room with the least number to be prime and then went on to number his rooms accordingly. Needless to say, the hotel was a phenomenal project. The mathematicians who visited the project were mighty impressed and recommended it to all their friends. With mathematicians from all over the world making it a point to stay here at least once, demand went through the roof. Having said that, all the mathematicians wanted to stay in prime number rooms and so occupancy never crossed 40%. Xuomi went out of business and sold his property to Mr. Frump who made millions by making it a casino.
What is the average of the room numbers in the first floor?
What is the largest room number?
What is the smallest room number?
What is the overall average of the 10 room numbers?
Mr. Goras had this to say when he visited the project. "By now you should know that nothing good comes out of listening to Pytha. If instead of listening to his daft idea of going for a sequence where the average was also prime (I mean, who goes for the average being prime), if you had shown some vision and thought of a sequence where extending to the third floor would have gotten you 2 more primes, it would have been great". If Xuomi had taken this approach, what would have been the average of all the room numbers?
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