Ta có: $1983^{4}$≡ 1 (mod 10)
⇒($1983^{3}$)^495 ≡ $1^{495}$ ≡ 1 (mod 10)
⇒$1983^{1980}$ ≡ 1 (mod 10)
mà $1983^{3}$≡ 27 (mod 10)
⇒$1983^{1980}$.$1983^{3}$ ≡ 1.27 ≡ 27 (mod 10)
⇒$1983^{1983}$ ≡ 27 (mod 10) (1)
Ta lại có: 1917^4 ≡ 1 (mod 10)
⇒$1917^{4}$^479 ≡ $1^{479}$ ≡ 1 (mod 10)
⇒ $1917^{1916}$ ≡ 1 (mod 10)
mà 1917 ≡ 7 (mod 10)
⇒ $1917^{1916}$.1917 ≡ 1.7 ≡ 7 (mod 10)
⇒ $1917^{1917}$ ≡ 7 (mod 10) (2)
Từ (1) và (2) ⇒ $1983^{1983}$ - $1917^{1917}$ ≡ 27-17 (mod 10)
≡ 10 (mod 10)
≡ 0 (mod 10)
Vậy: $1983^{1983}$ - $1917^{1917}$ ≡ 0 (mod 10)
hay: $1983^{1983}$ - $1917^{1917}$ chia hết cho 10