Question

Chứng minh với mọi n thuộc N :

a) 7⁴ⁿ - 1 chia hết cho 5

b) 2⁴ⁿ⁺¹+3 chia hết cho 5

Answer 1

Lời giải:

a. Ta có:

$7^4\equiv 1\pmod 5$

$\Rightarrow 7^{4n}\equiv 1^n\equiv 1\pmod 5$

$\Rightarrow 7^{4n}-1\equiv 0\pmod 5$

Hay $7^{4n}-1\vdots 5$

b.

$2^4\equiv 1\pmod 5$

$\Rightarrow 2^{4n+1}=2.2^{4n}\equiv 2.1^n\equiv 2\pmod 5$

$\Rightarrow 2^{4n+1}+3\equiv 2+3\equiv 5\equiv 0\pmod 5$

$\Rightarrow 2^{4n+1}+3\vdots 5$