Question

Cho các số nguyên a, b, c, d \(\ne\)0 thỏa mãn: \(ab=cd\) . CMR:

\(a^{2018}+b^{2018}+c^{2018}+d^{2018}\) là hợp số.

Answer 1

Gọi \(n=\left(a,c\right)\) \(\Rightarrow\left\{{}\begin{matrix}a=na_1\\c=nc_1\end{matrix}\right.\)

+ \(ab=cd\Rightarrow na_1b=nc_1d\)

\(\Rightarrow a_1b=c_1d\) (1)

\(\Rightarrow b⋮c_1\Rightarrow b=mc_1\)

Thay \(b=mc_1\) vào (1) ta có :

\(a_1mc_1=c_1d\Rightarrow d=ma_1\)

Do đó : \(a^{2018}+b^{2018}+c^{2018}+d^{2018}\)

\(=\left(na_1\right)^{2018}+\left(mc_1\right)^{2018}+\left(nc_1\right)^{2018}+\left(ma_1\right)^{2018}\)

\(=a_1^{2018}\left(m^{2018}+n^{2018}\right)+c_1^{2018}\left(m^{2018}+n^{2018}\right)\)

\(=\left(a_1^{2018}+c_1^{2018}\right)\left(m^{2018}+n^{2018}\right)\)

=> đpcm