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 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 - 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?

 See The Solution Submitted by K Sengupta Rating: 3.3333 (3 votes)

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 Hint | Comment 1 of 5

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
 Posted by K Sengupta on 2007-04-19 01:21:01

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