Solve: 1/x + 1/y =1/6
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1/X + 1/Y=1/6
Consider that whichever of 1/x or 1/y is bigger, it must be at least half the size of 1/6, or 1/12. If it weren’t, there’s no way you could add it to a smaller number to get 1/6. However, there are only 12 integer values for the bigger number that satisfy 1/a >= 1/12.
Thus there are only 24 possible solutions, 12 for 0<x<=12, and 12 for 0<y<=12. The question is also symmetrical for x and y, so you only have to test either set of 12, and then just swap x and y to find the rest. It isn’t particularly hard to check all possible values by hand and find the largest value for x.
In truth, you don’t even have to bother checking all of them if you don’t want to. If you want x to be as big as possible, you want 1/x to be as small as possible, but still greater than zero. This is equivalent to saying we want 1/6 - 1/y to be as small as possible if you rearrange the equation. The best way to do this is to have 1/y cancel out 1/6 as best as it can. y=6 seems like a good choice, but it means x=1/0, so that’s no good. The next closest option is y=7, which gives x=42. Any other choice of y will give a larger value of 1/x (or a negative, which is even worse), and thus a smaller value of x, so there’s your solution
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