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Unformatted text preview: Math 465 Solution Set 9 Ae Ja Yee 1. Prove that { 3 , 4 , 5 } is the only primitive Pythagorean triple involving consecutive pos itive integers. Solution. Let x = a 1 , y = a, z = a + 1. Then ( a 1) 2 + a 2 = ( a + 1) 2 2 a 2 2 a + 1 = a 2 + 2 a + 1 a 2 4 a = 0 a ( a 4) = 0 which gives that a = 0 or 4. Thus { 3 , 4 , 5 } is the only primitive Pythagorean triple involving consecutive positive integers. 2 2. Find all Pythagorean triangles whose areas are equal to their perimeters. (Hint: Solve x 2 + y 2 = z 2 and xy = 2( x + y + z ).) Solution. From xy = 2( x + y + z ), we get z = 1 2 xy x y So, x 2 + y 2 = x 2 y 2 4 + x 2 + y 2 x 2 y xy 2 + 2 xy x 2 y 2 4 x 2 y 4 xy 2 + 8 xy = 0 xy ( xy 4 x 4 y + 8) = 0 Since x, y are sides of a triangle, xy 6 = 0. So, xy 4 x 4 y + 8 = 0 ( x 4)( y 4) = 8 Then the solutions for x, y, z are x = 5 , y = 12 , z = 13 , x = 6 , y = 8 , z = 10 , x = 8 , y = 6 , z = 10 , x = 12 , y = 5 , z = 13 ....
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 Spring '08
 YEE
 Number Theory, Integers

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