Question

[Nguyễn Đình Vũ]

Cho A=2 + 2 mũ 2 + 2 mũ 3... + 2 mũ 60. Chứng tỏ rằng: A chia hết cho 3, 7 và 5

Answer 1

[Akai Haruma]

Lời giải:
$A=2+2^2+2^3+...+2^{60}$

$2A=2^2+2^3+2^4+...+2^{61}$

$\Rightarrow 2A-A=2^{61}-2$

Hay $A=2^{61}-2$

Ta thấy: 

$2\equiv -1\pmod 3$

$\Rightarrow 2^{61}\equiv (-1)^{61}\equiv -1\pmod 3$

$\Rightarrow A=2^{61}-2\equiv -1-2\equiv -3\equiv 0\pmod 3$

Vậy $A\vdots 3$

Mặt khác:
$2^3\equiv 1\pmod 7$

$\Rightarrow 2^{61}=(2^3)^{20}.2\equiv 1^{20}.2\equiv 2\pmod 7$

$\Rightarrow A=2^{61}-2\equiv 2-2\equiv 0\pmod 7$ 

Vậy $A\vdots 7$

Lại có:

$2^4\equiv 1\pmod 5$

$\Rightarrow 2^{61}=(2^4)^{15}.2\equiv 1^{15}.2\equiv 2\pmod 5$

$\Rightarrow A=2^{61}-2\equiv 2-2\equiv 0\pmod 5$

Vậy $A\vdots 5$ 

Ta có đpcm.