Ta co nⁿ - n² + n - 1 = nⁿ - n - (n² - 2n + 1)
= nⁿ - n - (n - 1)²
=> ta can chung minh nⁿ - n chia het cho (n - 1)²
Ta co nⁿ - n = n.[ n^(n - 1) - 1 ] = n.(n - 1).[ n^(n - 2) + n^(n - 3) + ... + n + 1 ]
Ta co n^(n - 2) + n^(n - 3) + ... + n + 1 dong du voi 1 + 1 + ... + 1 (n - 1 chu so 1) (mod n - 1)
=> n^(n - 2) + n^(n - 3) + ... + n + 1 dong du voi n - 1 (mod n-1)
=> n^(n - 2) + n^(n - 3) + ... + n + 1 = k(n - 1)
=> n.(n - 1).[ n^(n - 2) + n^(n - 3) + ... + n + 1 ] = kn(n - 1)²
=> nⁿ - n = kn(n - 1)² => nⁿ - n chia het cho (n - 1)²
Vậy (dpcm)
Chúc you hok tốt 👍👍👍👍