Question 1103484
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A formal algebraic solution to this problem is a good exercise.<br>
But there is a big shortcut you can take if you are just looking to get the answer as quickly as possible -- as, for example, if the question is on a competitive timed test.<br>
The difference between a 3-digit number and the number with the same digits reversed is always a multiple of 99.  Specifically, if the difference between the two numbers is 198 = 2*99, then the first and last digits of the original number differ by 2.<br>
In this problem, 198=2*99 is added to the original number to get the new number; that means the possibilities for the original number are 1?3, 2?4, 3?5, 4?6, etc.<br>
The additional given information that the middle digit is 1 more than the sum of the first and third digits limits the possible answers to 153, 274, and 395.<br>
Finally the requirement that the original number must be equal to 17 times the sum of the digits shows that the original number has to be 153.