Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
Already signed up?
New to KöMaL?
I want the old design back!!! :-)

Problem C. 1111. (February 2012)

C. 1111. The sum of all edges of two cubes is divisible by 72. Prove that the sum of their volumes is divisible by 6.

(5 pont)

Deadline expired on March 12, 2012.


Sorry, the solution is available only in Hungarian. Google translation

Megoldás. A kockák éleinek hossza legyen \(\displaystyle n\) és \(\displaystyle m\). Ekkor az összes élük hossza \(\displaystyle E=12m +12n\) osztható 72-vel, ami szerint \(\displaystyle n+m\) osztható 6-tal. Térfogatösszegük \(\displaystyle S=m^3+n^3=(m+n)(m^2-mn+n^2)\): mivel olyan szorzat, melynek egyik tényezője 6 többszöröse, ezért \(\displaystyle S\) is osztható 6-tal.


Statistics:

344 students sent a solution.
5 points:61 students.
4 points:228 students.
3 points:11 students.
2 points:23 students.
1 point:12 students.
0 point:3 students.
Unfair, not evaluated:6 solutions.

Problems in Mathematics of KöMaL, February 2012