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Rex and Ralph: Mystery Number
Category:  Math 
Submitted By:  javaguru 
Fun:  (2.65) 
Difficulty:  (3.26) 
Two mathematicians, Rex and Ralph, have an ongoing competition to stump each other. Ralph was impressed by the ingenuity of Rex's last attempt using clues involving prime numbers, but he thinks he's got an even better one for Rex. He tells Rex he's thinking of a 6digit number.
"All of the digits are different. The digital sum matches the number formed by the last two digits in the number. The sum of the first two digits is the same as the sum of the last two digits."
"Take the sum of the number, the number rotated one to the left, the number rotated one to the right, the number with the first three and last three digits swapped, the number with the digit pairs rotated to the left, and the number with the digit pairs rotated to the right. The first and last digits of this sum match the last two digits of the number, in some order."
Ralph then asks, "If each of the three numbers formed by the digit pairs in the number is prime, then what is the number?"
Rex looks confused, and for a moment Ralph thinks he's finally gotten him. Then Rex smiles, scribbles a few things down on a pad of paper and then says, "Very nice, Ralph!"
Rex then tells Ralph his number.
What did Rex say?
(See the hint for an explanation of the terminology.)
Hint:
The digital sum is the sum of the digits in the number. The digital sum of 247 is 2+4+7 = 13.
The digit pairs in 125690 are 12 56 90. These are also the numbers formed by the digit pairs.
Rotating 123456 one to the left gives 234561;
Rotating 123456 one to the right gives 612345;
Rotating the digit pairs in 567890 to the left gives 789056;
Rotating the digit pairs in 567890 to the right gives 905678.
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