Question

Aratati ca 7^2+7^3+...+7^2017 divizibil cu 133

Answer 1

7^2+7^3+...+7^2017 are 2017-2+1 termeni, adica 2016. 2016 este divizibil cu 3, deci putem grupa termenii cate 3 astfel:
(7^2+7^3+7^4+...+(7^2015+7^2016+7^2017)=
=7(7+7^2+7^3)+...+7^2015(7+7^2+7^3)=
=7x399+(7^4)x399+...+(7^2015)x399=
=399x(7+7^4+7^7+...+7^2015)=
=3x133x(7+7^4+7^7+...+7^2015)=m133, m=multiplu
7^2+7^3+...+7^2017=m13=>ca se divide cu 133