Problem K. 311. (November 2011)

K. 311. The number 77²⁰⁰⁹+77²⁰¹⁰+77²⁰¹¹+77²⁰¹² is evidently divisible by 7 and 11. Find all prime factors of the number.

(6 pont)

Deadline expired on December 12, 2011.

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

Megoldás. \(\displaystyle 77^{2009}+77^{2010}+77^{2011}+77^{2012}=77^{2009}\left( 1+77+77^2 +77^3 \right)=7^{2009}\cdot 11^{2009}\cdot 462540= 7^{2009}\cdot 11^{2009}\cdot 2^2\cdot 3 \cdot 5 \cdot 13 \cdot 593\).

A feladatbeli kifejezés a következő prímszámokkal osztható: 2, 3, 5, 7, 11, 13, 593.

Statistics:

183 students sent a solution.
6 points:	107 students.
5 points:	22 students.
4 points:	8 students.
3 points:	10 students.
2 points:	6 students.
1 point:	15 students.
0 point:	12 students.
Unfair, not evaluated:	3 solutionss.

Problems in Mathematics of KöMaL, November 2011