Question

aratai ca numaruln=7+7²+7³+....+7²³+7 la puterea 24 este divizibil cu:
A)7 b)8. C)3
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Answer 1

n=7+7²+7³+....+7²³+7²⁴=7×(7⁰+7¹+7²+...+7²³+7²⁴)  deci divizibil cu 7
n=7+7²+7³+....+7²³+7²⁴=7×(7⁰+7¹)+7³(7⁰+7¹)²+...+7²³(7⁰+7¹)=
=7×(1+7)+7³(1+7)+...+7²³×(1+7)=8×(7+7³+...+7²³)   deci divizibil cu 8
n=7+7²+7³+....+7²³+7²⁴=
avem 24  termeni ii putem grupa cate 3
=(7+7²+7³)+....+7²¹×(7+7²+7³)=
=(7+49+343)+...+7²¹×(7+49+343)=
=399×(1+...+7²¹)=
=3×133×(1+...+7²¹) deci divizibil cu 3