Question

If $(27{)}^{999}$ is divided by 7, then the remainder is :

Answer 1

$(27{)}^{999}=(28-1{)}^{999}$

$={}^{999}{C}_0(28{)}^{999}-{}^{999}{C}_1(28{)}^{998}\dots .+{}^{999}{C}_{998}\left(28\right)-{}^{999}{C}_{999}$

$=28\left[{}^{999}{C}_0\right(28{)}^{998}-{}^{999}{C}_1(28{)}^{997}\dots \dots +{}^{999}{C}_{998}]-{}^{999}{C}_{999}$

$(27{)}^{999}=\frac{28\left[{}^{999}{C}_0\right(28{)}^{998}-{}^{999}{C}_1(28{)}^{997}\dots \dots .+{}^{999}{C}_{998}]}{7}-\frac{1}{7}$

So remainder is remainder of $\frac{-1}{7}$

which is $6$ (because $-1=7(-1)+6$)

(remainder is always positive so we will not consider $-1$).