Question

Chứng minh rằng \(^{31^{10}}\)-1\(⋮\)300

Answer 1

\(31^{10}-1=\left(31^2\right)^5-1^5=\left(31^2-1\right)\left(31^8+5.31^6+...+5.31+1\right)\\ =960.\left(31^8+5.31^6+...+5.31+1\right)\)

ta có: \(\left\{{}\begin{matrix}960⋮6\\960⋮10\\960⋮5\end{matrix}\right.\)

đồng thời : \(UCLN\left(6,10,5\right)=1\)

do đó:

\(960⋮\left(5.6.10\right)\: hay\: 960⋮300\\ \Rightarrow960.\left(31^8+5.31^6+...+5.31+1\right)⋮300\\ hay\: 31^{10}-1⋮300\left(đpcm\right)\)