Raise to the seventh power, get factor (Posted on 2007-04-18)

Determine at least three pairs of positive integers (x,y) with x< y such that xy(x+y) is not divisible by 7, but (x+y)^{7} - x^{7} - y^{7} is divisible by 7^{7}

Does the given problem generate an infinite number of pairs as solutions?

Can you do this in a short time using pen and paper, and eventually a hand calculator, but no computer programs?