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Problem C. 1160. (March 2013)

C. 1160. What is the remainder if the sum 20122013+20132012 is divided by 2012.2013?

Suggested by D. Fülöp, Pécs

(5 pont)

Deadline expired on April 10, 2013.


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

Megoldás. Vizsgáljuk a következő törtet:

\(\displaystyle t=\frac{2012^{2013}+2013^{2012}}{2012\cdot2013}=\frac{2012^{2012}}{2013}+\frac{2013^{2011}}{2012}.\)

Használjuk fel a következőket:

\(\displaystyle 2012^{2012}=(2013-1)^{2012}=2013^{2012}-2012\cdot2013^{2011}+\binom{2012}{2}\cdot2013^{2010}-...-\binom{2012}{2011}\cdot2013+\binom{2012}{2012}\cdot1=\)

\(\displaystyle =2013a+1,~{\rm{ahol}}~a\in \Bbb N^+;\)

\(\displaystyle 2013^{2011}=(2012+1)^{2011}=2012^{2011}+2011\cdot2012^{2010}+\binom{2011}{2}\cdot2012^{2009}+...+\binom{2011}{2010}\cdot2012+\binom{2011}{2011}\cdot1=\)

\(\displaystyle =2012b+1,~{\rm{ahol}}~b\in \Bbb N^+.\)

Ezekből

\(\displaystyle t=\frac{2013a+1}{2013}+\frac{2012b+1}{2012}=a+\frac{1}{2013}+b+\frac{1}{2012}=a+b+\frac{2012+2013}{2012\cdot2013}=a+b+\frac{4025}{2012\cdot2013}.\)

Vagyis a keresett maradék 4025.


Statistics:

157 students sent a solution.
5 points:52 students.
4 points:18 students.
3 points:37 students.
2 points:18 students.
1 point:11 students.
0 point:20 students.
Unfair, not evaluated:1 solution.

Problems in Mathematics of KöMaL, March 2013