MathDB

a) Determine all 4-tuples

(x_0,x_1,x_2,x_3)

of pairwise distinct intergers such that each

x_k

is coprime to

x_{k+1}

(indices reduces modulo 4) and the cyclic sum

\frac{x_0}{x_1}+\frac{x_1}{x_2}+\frac{x_2}{x_3}+\frac{x_3}{x_1}

is an interger. b)Show that there are infinitely many 5-tuples

(x_0,x_1,x_2,x_3,x_4)

of pairwise distinct intergers such that each

x_k

is coprime to

x_{k+1}

(indices reduces modulo 5) and the cyclic sum

\frac{x_0}{x_1}+\frac{x_1}{x_2}+\frac{x_2}{x_3}+\frac{x_3}{x_4}+\frac{x_4}{x_0}

is an interger.

Problem Statement