Given positive integers x and y such that x does not equal y and 1/x + 1/y = 1/28 , what is the smallest possible value for x+y ?
x, y must be > 28
Let z = 28
Let x = z + a
Let y = z + b so we have
1/ (z + a) + 1/ ( z + b) = 1/z
(z + b + z + a) / ( z^2 + az + bz + ab) = 1/z
( 2z + a + b) / ( z^2 + az + bz + ab) = 1/z cross-multiply
z( 2z + a + b) = z^2 + az +bz + ab
2z^2 + az + bz = z^2 + az + bz + ab
z^2 = ab
28^2 = ab
784 = ab
Factors of 784 = ± [1,2,4,7,8,14,16,28,49,56, 98,112,196, 392,784]
ab = 784*1 or (-784) (-1)
Largest value of a + b = 784 + 1
Smallest value of a + b = -784 - 1
x = z + a = 28 + (-784) = -756
y = z + b = 28 + (-1) = 27
Smallest value of x + y = -756 + 27 = -729