Giải thích các bước giải:
Ta có:
$9$ lẻ $\to 9^{2012}$ lẻ
$3$ lẻ $\to 3^{43}$ lẻ
$8$ chẵn $\to 8^{30}$ chẵn
$\to 9^{2012}-3^{43}-8^{30}$ chẵn
$\to 9^{2012}-3^{43}-8^{30}\quad\vdots\quad 2(*)$
Ta có:
$9\equiv -1(mod 5)$
$\to 9^{2012}\equiv (-1)^{2012}\equiv 1(mod 5)$
Mặt khác:
$9\equiv -1(mod 5)$
$\to 3^2\equiv -1(mod 5)$
$\to (3^2)^{21}\equiv (-1)^{21}\equiv -1(mod 5)$
$\to 3^{42}\equiv -1(mod 5)$
$\to 3^{42}\cdot 3\equiv -3(mod 5)$
$\to 3^{43}\equiv -3(mod 5)$
Ta có:
$4\equiv -1(mod 5)$
$\to 4^{44}\equiv (-1)^{44}\equiv 1(mod 5)$
$\to (2^2)^{44}\equiv 1(mod 5)$
$\to 2^{88}\equiv 1(mod 5)$
$\to 2^{88}\cdot 2^2\equiv 2^2\equiv -1(mod 5)$
$\to 2^{90}\equiv -1(mod 5)$
$\to (2^3)^{30}\equiv -1(mod 5)$
$\to 8^{30}\equiv -1(mod 5)$
$\to 9^{2012}-3^{43}-8^{30}\equiv 1-(-3)-(-1)\equiv 5\equiv 0(mod 5)$
$\to 9^{2012}-3^{43}-8^{30}\quad\vdots\quad 5(**)$
Vì $(2,5)=1\to$Từ $(*),(**)$
$\to 9^{2012}-3^{43}-8^{30}\quad\vdots\quad 2\cdot 5$
$\to 9^{2012}-3^{43}-8^{30}\quad\vdots\quad 10$
