Local Mathematical Competitions

Question

Of the vertices of a cube, 7 of them have assigned the value 0, and the eighth the value 1. A *move* is selecting an edge and increasing the numbers at its ends by an integer value k > 0. Prove that after any finite number of moves, the g.c.d. of the 8 numbers at vertices is equal to 1.

Accepted Answer

Let us alternately colour the vertices in black and white. After any move, the difference between the sums of the numbers at the black and the white vertices remains 1, therefore the g.c.d. of the 8 numbers is equal to 1 (as it is dividing the difference of the sums mentioned above).