Question

The difference between two positive integers is 2 and the difference between their cubes is 56. find the number

Answer 1

let the two numbers be x and y then x - y = 2 and x^2 + y^2 - 2xy = 4

and x^3 - y^3 = 56 but (x^3 - y^3) = (x - y) (x^2 + y^2 + xy) = 56

substituting x - y = 2 we get (x^2 + y^2 + xy) = 28 but x^2 + y^2 = 4 + 2xy

substituting we get 3xy + 4 = 28 that implies xy = 8

if difference of two numbers is 2 and their product is 8 then the numbers must be 4 and 2.