Given positive integers x and y such that x is unequal to y and 1/x+1/y=1/18, what is the smallest possible value for x+y?
1/x + 1/y = 1/18
Let z = 18
And x, y must be > 18
So
Let x = z + a and Let y = z + b
So we have that
1 1 1
____ + ____ = _____ simplify
z + a z + b z
[ z + b + z + a] 1
____________ = _______
[ z + a] [z + b] z
[2z + a + b] 1
_____________ = _____ cross-multiply
[ z+ a ] [ z + b] z
z [ 2z + a + b ] = [ z + a] [ z + b ]
2z^2 +az + bz = [ z + a ] [ z + b ]
2z^2 + az + bz = z^2 + az + bz + ab
z^2 = ab
(18)^2 = ab
324 = ab
The factors of 324 are
1 | 2 | 3 | 4 | 6 | 9 | 12 | 18 | 27 | 36 | 54 | 81 | 108 | 162 | 324
The smallest possible value for a,b is 12 , 27
So
x = z + a = 18 + 12 = 30
y = z + b = 18 + 27 = 45
So....the minimum value for x + y = 75