Termenii din suma pot fi grupati 2 cate 2
[tex]1+6=7[/tex]
[tex]6^{2}+6^{3}=6^{2}(1+6)=7*6^{2}[/tex]
[tex]6^{4}+6^{5}=6^{4}(1+6)=7*6^{4}[/tex]
-----------------------------------------------------
[tex]6^{100}+6^{101}=6^{100}(1+6)=7*6^{100}[/tex]
Acum le adunam pe toate
[tex]a=1+6+6^{2}+6^{3}+..+6^{100}+6^{101}=7+7*6^{2}+7*6^{4}+..+7^{100}=7(1+6^{2}+6^{4}+..+6^{100})=7b[/tex]
notam acea suma cu b
Grupam din nou termenii 2 cate 2
[tex]1+6^{2}=1+36=37[/tex]
[tex]6^{4}+6^{6}=6^{4}(1+36)=37*6^{4}[/tex]
-----------------------------------------------------------
[tex]6^{98}+6^{100}=6^{98}(1+36)=37*6^{98}[/tex]
Adunam toti termenii
[tex]b=1+6^{2}+6^{4}+6^{6}+6^{8}+..+6^{98}+6^{100}=37+37*6^{4}+..+37*6^{98}=37(1+6^{4}+..+6^{98})=37*c[/tex]
Am notat cu c suma ramasa
Acum grupam termenii 3 cate 3
[tex]1+6^{4}+6^{8}=1680913[/tex]
[tex]6^{12}+6^{16}+6^{20}=6^{12}(1+6^{4}+6^{8})=1680913*6^{12}[/tex]
---------------------------------
[tex]6^{90}+6^{94}+6^{98}=6^{90}(1+6^{4}+6^{8})=1680913*6^{90}[/tex]
Atunci adunam toti termenii
[tex]c=1+6+6^{4}+6^{8}+..+6^{98}=1680913+1680913*6^{10}+..+1680913*6^{90}=1680913(1+6^{12}+..+6^{90})=1680913d[/tex]
Numarul acela este divizibil cu 43
1680813=43*39091
Atunci
[tex]c=43*39091*d[/tex]
Si atunci rezulta ca
[tex]a=7*37*43*39091*d[/tex] care e divizibil atunci cu 37*7*43
