Question

Arătați că numărul $13 ^{997} +13 ^{998} +13 ^{999} +......13 ^{2016}$ se divide cu 427

Answer 1

427=7*61
trebuie sa demonstram ca e divizibil cu 7 si cu 61
nr de termeni 2015-997+1=1020  (multiplu de 2)
ii putem grupa cate 2
13^997+13^998+13^999+......+13^2014+13^2015+13^2016=
=13^997(13^0+13^1)+............+13^2015(13^0+13^1)=
=13^997(1+13)+............+13^2015(1+13)=
=14×(13^997+............+13^2015)=
=2×7×(13^997+............+13^2015)= deci divizibil cu7

nr de termeni 2015-997+1=1020     (multiplu de 3)
ii putem grupa cate 3
13^997+13^998+13^999+......+13^2014+13^2015+13^2016=
=13^997(13^0+13^1+13^2)+......+13^2014(13^0+13^1+13^2)=
=13^997(1+13+169)+......+13^2014(1+13+169)=
=183 ×(13^997+......+13^2014)=
=3×61 ×(13^997+......+13^2014) divizibil cu 61