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 - x7 - y7 is divisible by 77
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?
Remember that xy(x+y) is NOT divisible by 7, but:
(x+y)^7 - x^7 - y^7 IS divisible by 7^7.
(i) Start by noting the identity:
(x + y)^7 - x^7 - y^7 = 7xy(x + y)(x^2 + xy + y^2)^2.
(What does this yield?)
(ii) Apply Euler Fermat Theorem.
Edited on April 19, 2007, 1:24 am