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A 2547 Puzzle (Posted on 2007-03-29) Difficulty: 3 of 5
a and b are positive integers. Dividing a2 + b2 by a + b we obtain the quotient as q and the remainder as r.

Determine analytically all possible pairs (a, b) such that q2 + r = 2547

  Submitted by K Sengupta    
Rating: 2.0000 (2 votes)
Solution: (Hide)
a^2 + b^2 >= (a + b)^2/2, so q >= (a + b)/2. Hence r < 2q.

The largest square less than 2547 is 2500 = 50^2 and:
2547 = 50^2 + 47.

The next largest gives 2547 = 49^2 + 146 . But 146 > 2*43.

So we must have q = 50, r = 47.
Hence a^2 + b^2 = 50(a + b) + 47
So, (a - 25)^2 + (b - 25)^2 = 1297 = 36^2 + 1^2

Accordingly, (a-25, b-25) = (36, 1); (36, -1); (1, 36); (-1, 36), since both a and b are positive integers.

Consequently, (a,b) = (61, 26); (61, 24); (26,61); (24, 61)

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Questionpositive- yes integers-noAdy TZIDON2007-03-31 11:34:06
Some Thoughtsre: My startAdy TZIDON2007-03-31 10:43:18
My startJer2007-03-30 11:30:41
Some Thoughtsa startDej Mar2007-03-30 00:54:54
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