[กิตติ]

2.$5(5!)^{2000}$ $(120)^{2000}$
แยกเป็น $8^{2000}\times 3^{2000} \times 5^{2000}$
$8^2 \equiv 64 (mod 7) \equiv 1(mod7)$
$2000 =2(1000)+0$
$8^{2000} \equiv 8^0(mod 7) \equiv 1(mod7)$...เศษ 1

$3^2 \equiv 9 (mod 7) \equiv 2(mod7)$
$3^3 \equiv 6 (mod 7) $
$3^4 \equiv 18 (mod 7) \equiv 4(mod7)$
$3^5 \equiv 12 (mod 7) \equiv 5(mod7)$
$3^6 \equiv 15 (mod 7) \equiv 1(mod7)$
$3^6 \equiv 1(mod7)$
$2000=6(333)+2$
$3^{2000} \equiv 9 (mod 7) \equiv 2(mod7)$......เศษ 2

$5^2 \equiv 25 (mod 7) \equiv 4(mod7)$
$5^3 \equiv 20 (mod 7) \equiv 6(mod7)$
$5^4 \equiv 30 (mod 7) \equiv 2(mod7)$
$5^5 \equiv 10 (mod 7) \equiv 3(mod7)$
$5^6 \equiv 15 (mod 7) \equiv 1(mod7)$
$2000=6(333)+2$
$5^{2000} \equiv 25 (mod 7) \equiv 4(mod7)$......เศษ 4

$8^{2000} \equiv 1(mod7)$ $3^{2000} \equiv 2(mod7)$
$8^{2000} \times 3^{2000} \equiv 1\times 2(mod7) \equiv 2(mod7)$
$8^{2000} \times 3^{2000} \equiv 2(mod7)$ กับ $5^{2000} \equiv 4(mod7)$
$8^{2000} \times 3^{2000} \times 5^{2000}\equiv 2\times 4(mod7) \equiv 1(mod7)$
$(120)^{2000}\equiv 1(mod7)$
$5\times (120)^{2000}\equiv 5\times 1(mod7) \equiv 5(mod7)$
ตอบเศษ 5

ให้ง่ายอีกหน่อยเมื่อกี้รู้แล้วว่า$(4!)^{2000} \equiv 2(mod7)$
$5^{2000} \equiv 4(mod7)$
$(4!)^{2000}\times5^{2000} \equiv 2\times 4(mod7) \equiv 1(mod7)$
$(5!)^{2000} \times 5 \equiv 5(mod7)$