Paper 4, Section I, D

Numbers and Sets | Part IA, 2017

(a) Show that for all positive integers zz and nn, either z2n0(mod3)z^{2 n} \equiv 0(\bmod 3) or z2n1(mod3)z^{2 n} \equiv 1(\bmod 3).

(b) If the positive integers x,y,zx, y, z satisfy x2+y2=z2x^{2}+y^{2}=z^{2}, show that at least one of xx and yy must be divisible by 3 . Can both xx and yy be odd?

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