let x and x+1 are the two consecutive integers, x is integer

The sum of the reciprocals of two consecutive integers is 17/72:

1/x + 1/(x + 1) = 17/72

by solving we find:

x = 8

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indeed:

1/8 + 1/9 = (9 + 8)/72 = 17/72

the two integers are 8 and 9.