+166 Given positive integers x and y such that x doesn't equal y and 1/x+1/y=1/12, what is the smallest possible value for x+y?
+4622 We have \(\frac{x+y}{xy}=\frac{1}{12}, 12(x+y)=xy\) . \(12x+12y=xy\) . Try to plug in numbers!
+129899 1/x + 1/y = 1/12
x + y = xy/12
12(x + y) = xy
12x - xy = -12y
x (12 - y) = -12y
x ( y - 12) = 12y
x = 12y / ( y - 12)
The smallest that y can be and still have x positive is when y = 13
(1) If y = 13 x = 156
(2) If y =14 x = 84
3) If y = 15 x = 60
(4) If y = 16 x = 48
(5) If y = 18 x = 36
(6) If y = 20 x = 30
(7) If y = 24 x = 24 but x and y cannot be the same
(8) If x = 30 y = 20
And every other combo will be a repeat of the sum of x and y for (1) - (6)
So.....the smallest value of x + y = 50
