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presidentabrahamlincoln

Show an algebraic expression is divisible by 4.

Question

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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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.

 

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