58th Ukrainian National Mathematical Olympiad

Question

Let [x_1, x_2] and [y_1, y_2, y_3] be the least common multiple of x_1, x_2 and y_1, y_2, y_3 respectively. For any positive integers a, b, c, d let A and B be such that: A = [a, b, c] [a, b, d] [a, c, d] [b, c, d] and B = [a, b] [a, c] [a, d] [b, c] [b, d] [c, d]. Show that A^6 B^4.

Accepted Answer

Take prime p that divides abcd. Without loss of generality, let the prime number be a factor of a, b, c, d of degree a_1 b_1 c_1 d_1 respectively. We can find the biggest degree of p—p_A and p_B, that divide A and B respectively. p_A = 3a_1 + b_1, p_B = 3a_1 + 2b_1 + c_1. Then degrees p'_A and p'_B for numbers A^6 and B^4 equal: p'_A = 6p_A = 18a_1 + 6b_1 p'_B = 4p_B = 12a_1 + 8b_1 + 4c_1.