Question

Va rog, problema 18......

Answer 1

$Pentru~olimpiada~trebuie~cunoscute~doua~proprietati~importante: \\ \\ (a+b)^n=M_a+b^n~(M_a=multiplu~de~a~;~a,b,n \in N) ~si~\\ \\ a^n+b^n~ \vdots~(a+b) \Leftrightarrow~n=impar. \\ \\ Sa~le~punem~in~practica! \\ \\ a)~8^{1342}=2^{4026}=4^{2013}. \\ \\ 37^{2013}+8^{1342}=37^{2013}+4^{2013}~care~este~divizibil~cu~(37+4)=41,~ \\ \\ conform~celei~de~a~doua~proprietati.$

$b)~22^{42}=(9 \cdot 2+4)^{42}=M_{9}+4^{42}=M_{9}+2^{84}. \\ \\ 22^{42}+7^{83}-2^{83}=M_9+(2^{84}-2^{83})+7^{83}=M_{9}+2^{83}+7^{83}. \\ \\ Proprietatea~2~ne~spune~ca~2^{83}+7^{83}~ \vdots~(2+7),~si~cum~M_9 ~ \vdots~9~(fiind~ \\ \\ un~multiplu~de~9)~rezulta~concluzia.$

$Edit:~Iar~(a-b)^n=a^n+(-b)^n.$