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Square pairs (Posted on 2005-06-15) Difficulty: 2 of 5
There are pairs of numbers whose sum and product are perfect squares. For instance, 5 + 20 = 25 and 5 x 20 = 100.

If the smallest number of such a pair is 1090, what is the smallest possible value of the other number? No computers!!

  Submitted by pcbouhid    
Rating: 2.6667 (3 votes)
Solution: (Hide)
Congratulations to all three, specially to Paul and Dickson, that proved the number (solution) achieved by Lisa, to be the minimum.

We have :

1090 + X = A^2

1090 * X = B^2

Since 1090 has no square factor > 1, 1090 must be (by the second equation above) a factor of X; in fact, X must be of the form 1090*C^2.

Thus, A^2 = 1090 + 1090 * C^2 = 1090*(C^2 + 1).

By similar reasoning, 1090 is a factor of (C^2 + 1). The smallest value of C^2 wich makes (C^2 + 1) divisible by 1090 is 1089, wich, fortunately, happens to be a perfect square, i.e., 33^2.

Hence, X = 1090 * 1089 = 1,187,010.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionPuzzle SolutionK Sengupta2008-11-21 23:49:04
answerK Sengupta2008-11-21 00:39:48
to juannickson2005-08-02 09:02:41
re: here is the solutionjuan2005-07-06 18:13:38
Solutionhere is the solutionnickson2005-06-17 10:05:56
Hints/Tipsproof that previous solution is minimalPaul2005-06-15 21:09:10
SolutionSolutionLisa2005-06-15 12:58:50
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