Question 703375
Note that since the integers are positive and consecutive, let them be represented by x and (x + 1). 

Reciprocal of x => 1/x

Reciprocal of (x + 1) => 1/(x + 1)

1/x + 1/(x + 1) = 1772

Getting a common denominator of x(x + 1):

(x + 1) + x = 1772(x)(x + 1)

2x + 1 = 1772x^2 + 1772x

1772x^2 + 1770x - 1 = 0

Via the quadratic formula:

x = (-1770 +/- √(1770^2 - 4(1772)(-1))/(2*1772) yields no integers at all. Pleas recheck your problem.