Question

Chứng minh rằng: \(-0,7\left(43^{43}-17^{17}\right)\) là một số nguyên.

Answer 1

Ta có:

\(43^{43}=43^{40}.43^3=\left(43^4\right)^{10}.43^3\)

\(=\left(...1\right)^{10}.\left(...7\right)=\left(...1\right).\left(...7\right)=\left(...7\right)\left(1\right)\)

Lại có:

\(17^{17}=17^{16}.17^1=\left(17^4\right)^4.17\)

\(=\left(...1\right)^4.\left(...7\right)=\left(...1\right).\left(...7\right)=\left(...7\right)\left(2\right)\)

Từ \(\left(1\right)\) và \(\left(2\right)\)

\(\Rightarrow-0,7\left(43^{43}-17^{17}\right)=-0,7\left(...7-...7\right)\)

\(=-0,7.\left(...0\right)\)

Mà: \(\left\{{}\begin{matrix}-0,7\in Z\\\left(...0\right)\in Z\end{matrix}\right.\)\(\Rightarrow-0,7.\left(...0\right)\in Z\)

Vậy \(-0,7\left(43^{43}-17^{17}\right)\) là một số nguyên (Đpcm)