by Chứng minh rằng:
\(a)\ P={{7}^{19}}+{{7}^{20}}+{{7}^{21}}\) chia hết cho \(57\)
\(b)\ Q={{32}^{567}}-{{32}^{566}}\) chia hết cho \(31\)
Lời giải chi tiết:
\(a)\ P={{7}^{19}}+{{7}^{20}}+{{7}^{21}} \\ ={{7}^{19}}+{{7}^{19}}.7+{{7}^{19}}{{.7}^{2}} \\ ={{7}^{19}}.\left( 1+7+{{7}^{2}} \right)\\ ={{7}^{19}}\left( 1+7+49 \right) \\={{7}^{19}}.57\)
Suy ra \(P\) chia hết cho \(57\).
\(b)\ Q={{32}^{567}}-{{32}^{566}} \\ ={{32}^{566}}.32-{{32}^{566}}.1 \\ ={{32}^{566}}\left( 32-1 \right) \\ ={{32}^{566}}.31\)
Suy ra \(Q\) chia hết cho \(31\).
