`1930^1930 + 1945^1945 + 1954^1954 + 1975^1975 - 2011^2011`

`= (...0) + (...5) + (1954^2)^977 + (...5) - (...1)`

`= (...5) + (...6)^977 + (...5) - (...1)`

`= (...5) + (...6) + (...5) - (...1)`

`= (...1) + (...5) - (...1)`

`= (...5) \vdots 5`

Vậy `1930^1930 + 1945^1945 + 1954^1954 + 1975^1975 - 2011^2011 \vdots 5`

`#dtkc`

`1930^1930 vdots 5`

`1945^1945 vdots 5`

`1975^1975 vdots 5`

Ta cần chứng minh : `1954^1954 - 2011^2011 vdots 5`

`-` Cách `1`

`1954^1954 - 2011^2011`

`= (1954^2)^977 - overline{(...1)}`

`= overline{(...6)}^977 - overline{(...1)}`

`= overline{(...6)} - overline{(...1)}`

`= overline{(...5)}`

`=> 1954^1954 - 2011^2011 vdots 5`

`=> 1930^1930 + 1945^1945 + 1954^1954 + 1975^1975 - 2011^2011 vdots 5`

`-` Cách `2`

`1954 -= -1 (mod 5)`

`=> 1954^1954 -= (-1)^1954 = 1(mod 5)`

`2011 -= 1(mod 5)`

`=> 2011^2011 -= 1^2011 = 1(mod 5)`

`=> 1954^1954 - 2011^2011 -= 1 - 1 -= 0 (mod 5)`

`=> 1954^1954 - 2011^2011 vdots 5`

`=> 1930^1930 + 1945^1945 + 1954^1954 + 1975^1975 - 2011^2011 vdots 5`

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