presidentabrahamlincoln 0 Posted February 27, 2017 Report Share Posted February 27, 2017 Let w, x, y, and z belong to the complete set of integers. If each of wxy + z^2, wxz + y^2, wyz + x^2, xyz + w^2 is divisible by 4, show that w^3 + x^3 + y^3 + z^3 is divisible by 4. Quote Link to post Share on other sites

0 plasmid 41 Posted March 4, 2017 Report Share Posted March 4, 2017 Modular arithmetic to the rescue Spoiler wxy + z^2 = 0 mod 4 wxz + y^2 = 0 mod 4 wyz + x^2 = 0 mod 4 xyz + w^2 = 0 mod 4 Multiply both sides of each of those equations by the squared term to get wxyz + z^3 = 0 mod 4 wxyz + y^3 = 0 mod 4 wxyz + x^3 = 0 mod 4 wxyz + w^3 = 0 mod 4 Add all four of those equations together 4*wxyz + w^3 + x^3 + y^3 + z^3 = 0 mod 4 Since 4*wxyz is divisible by 4, that means w^3 + x^3 + y^3 + z^3 must also be divisible by 4. Quote Link to post Share on other sites

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## presidentabrahamlincoln 0

Let w, x, y, and z belong to the complete set of integers.

If each of wxy + z^2, wxz + y^2, wyz + x^2, xyz + w^2 is divisible by 4, show that

w^3 + x^3 + y^3 + z^3 is divisible by 4.

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