CMAT 2014 Solved Question Paper

(a) Rem $\frac{{888}^{222}+{222}^{888}}{5}$ = Rem$\frac{{888}^{222}}{5}$ + Rem$\frac{{222}^{888}}{5}$
=Rem $\frac{3^{222}}{5}$ + Rem$\frac{2^{888}}{5}$
$\left\{\begin{array}{c}\because \frac{222}{5}=44\frac{2}{5}\\ \mathit{a}\mathit{n}\mathit{d}\frac{888}{5}=177\frac{3}{5}\end{array}\right\}$
= Rem$\frac{{\left(3^4\right)}^{55}\times 3^2}{5}$ + Rem$\frac{{\left(2^4\right)}^{222}}{5}$
= Rem$\frac{1\times 9}{5}$ + Rem$\frac{1}{5}$ = Rem$\frac{4}{5}$ + Rem$\frac{1}{5}$
= Rem$\frac{4+1}{5}$ = Rem$\frac{5}{5}$ = 0

$\mathbf{Shortcut}$
$8^{222}$ → 4 (unit digit)
$2^{888}$ → 6 (unit digit)
∴ 4 + 6 = 10 which is divisible by 5.
∴ Remainder = 0