Cho A=2+2 mũ 2+2 mũ 3+...+2 mũ 60.Chứng tỏ rằng: a,A chia hết cho 3 b,A chia hết cho7 c,Achia hết cho 5

Đáp án+Giải thích các bước giải:

$a)A=2+2 ^ 2+2 ^ 3+...+2 ^{60}\\=(2+2 ^ 2)+(2 ^ 3+2^4)+ . . . + (2^{59}+2 ^{60})\\=2.(1+2)+2^3.(1+2)+. . . +2^{59}.(1+2)\\=2.3+2^3.3+. . . +2^{59}.3\\=3.(2+2^4+.. . +2^{59})\vdots3\\Vậy \ A \ \vdots \ 3$

$b)A=2+2 ^ 2+2 ^ 3+...+2 ^{60}\\=(2+2 ^ 2+2 ^ 3)+...+(2^{58}+2^{59}+2 ^{60})\\=2.(1+2+4)+...+2^{58}.(1+2+4)\\=2.7+. ..+2^{58}.7\\=7.(2+. . +2^{58})\vdots 7\\Vậy\ A\ \vdots \ 7$

$c)A=2+2 ^ 2+2 ^ 3+...+2 ^{60}\\=(2+2^3)+(2^2+2^4)+. . .+(2^{58}+2^{60})\\=2.(1+4)+2^2.(1+4)+...+2^{58}.(1+4)\\=2.5+2^2.5+...+2^{58}.5\\=5.(2+2^2+... +2^{58})\vdots 5\\Vậy\ A\ \vdots \ 5$