Question

Chứng minh rằng \(33^{66}+77^{55}-2\text{ ⋮ 5 }\)

Answer 1

Ta có: 33² \(\equiv\) -1 (mod 5)

=> (33²)³³ \(\equiv\) (-1)³³ (mod 5)

=> 33⁶⁶ \(\equiv\) -1 (mod 5) (1)

Lại có: 77² \(\equiv\) -1 (mod 5)

=> (77²)²⁷ \(\equiv\) (-1)²⁷ (mod 5)

=> 77⁵⁴ \(\equiv\) -1 (mod 5)

=> 77⁵⁴.77 \(\equiv\) (-1).77 (mod 5)

=> 77⁵⁵ \(\equiv\) -77 \(\equiv\) -2 \(\equiv\) 3 (mod 5) (2)

Từ (1) và (2) => 33⁶⁶ + 77⁵⁵ \(\equiv\) -1 + 3 \(\equiv\) 2 (mod 5

=> 33⁶⁶ + 77⁵⁵ - 2 ⋮ 5