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Question

The number of ordered pairs (x,y) of positive integers satisfying the equation 1/x + 1/y = 1/20 is

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Solution

1x+1y=120 (x,y0)1x=120-1y1x=y-2020yx=20yy-20Therefore, for x to be positive, y>20Starting with y=21x=20(21)=420for y=22x=10(22)=220for y=23, x is not an integerfor y=24, x=5(24)=120for y=25, x=(4)25=100for y=26,27,29 x is not an integerfor y=28, x=70for y=30, x=60for y=31,32,33,34,35 x is not an integerfor y=36, x=45for y=37,38,39 x is not an integerfor y=40, x=40Since the equation is symmetric w.r.t x and y, the reverse will be true if we start with assuming x (as we did for y)So possible values of x (or y) are=21,22,24,25,28,30,36,40,45,60,70,100,120,220,420so total pairs possible=15

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