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Rex and Ralph: Prime Clues
Category:  Math 
Submitted By:  javaguru 
Fun:  (2.4) 
Difficulty:  (3.16) 
Two mathematicians, Rex and Ralph, have an ongoing competition to stump each other. Frustrated that Ralph was able to so easily figure out his last question, Rex is certain this one won't be so easy. He tells Ralph he's thinking of a number.
"The numbers one less and one more than the number are both the product of five prime numbers. The three numbers together have thirteen prime factors, all different. The sum of the prime factors of the number is 1400."
"OK," says Ralph after a moment. "That's probably enough information to find the number with a bruteforce search, is that what you expect me to do?"
"No, no, no," replies Rex. "I don't want you to do anything as inelegant as that. Here's some more information."
"The digital sum of each of the number's prime factors is prime, as is digital sum of the product of these sums. In fact, if you the reverse the digits in the product's digital sum you get a different prime number that is the digital sum of the number I'm thinking of."
Ralph takes out a pad of paper and starts jotting down some notes.
"The middle two digits in the number are its only prime digits and the number formed by the middle two digits is also prime. The number formed by the first three digits in the number is prime and its digital sum is also prime. In fact, the digital sum of the digital sum, and the digital sum of the digital sum of the digital sum of the first three numbers are also both prime."
"What number am I thinking of?" asks Rex.
Ralph jots down a few more notes and then does a couple of calculations on his calculator. He says, "OK, I know it's one of two numbers, but I don't want to factor these to figure out which one."
Rex frowns and says, "One of those numbers is divisible by 13."
Ralph smiles and tells Rex the number.
What were the two numbers and which one was the one Rex was thinking of?
Note: The "digital sum" is the sum of the digits in a number. For example, the digital sum of 247 = 2 + 4 + 7 = 13.
Hint:
You don't have to factor any numbers to find the answer, nor do you need to determine the primality of any numbers larger than 1000.
The problem is as much about logic as it is about math.
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