Mathematical and Physical Journal
for High Schools
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Problem A. 472. (January 2009)

A. 472. Call a finite sequence (p1(x),...,pk(x)) of polynomials with integer coefficients Euclidean if there exist polynomials q_1(x),\ldots,q_k(x) with integer coefficients such that d(x)=q1(x)p1(x)+...+qk(x)pk(x) is a common divisor of p1(x),...,pk(x), i.e. there are polynomials r_1(x),\ldots,r_k(x) with integer coefficients for which pi(x)=ri(x)d(x) for every 1\lei\lek. Prove that whenever p_1(x),\ldots,p_n(x) are polynomials with integer coefficients and every two of them form a Euclidean pair, then the sequence \big(p_1(x),\ldots,p_n(x)\big) is Euclidean as well.

(5 pont)

Deadline expired on February 16, 2009.


3 students sent a solution.
5 points:Backhausz Tibor, Nagy 235 János, Tomon István.

Problems in Mathematics of KöMaL, January 2009