the two consecutive positive integers can be written as:

x and x+1, where x is positive integer, x + 1 is the larger (since x+1 > x)

the product of two consecutive positive integers is 27 more than 5 times the larger integer:

x(x + 1) = 27 + 5(x + 1)

by solving we find:

x = 8

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8 + 1 = 9

the integers are 8 and 9.