Question

If $7^{103}$ is divided by $25$ then find the remainder.

Answer 1

$7^{103}=7\left(7^{102}\right)=7(7^2{)}^{51}=7(49{)}^{51}$
$=7(50-1{)}^{51}=7{[}^{51}{C}_0(50{)}^{51}{-}^{51}{C}_1\left(50{)}^{50}{+}^{51}{C}_2\right(50{)}^{49}-........{+}^{51}{C}_{50}\left(50\right){-}^{51}{C}_{51}\left(1\right)]$
$=7{[}^{51}{C}_0\left(50{)}^{51}{-}^{51}{C}_1\right(50{)}^{50}{+}^{51}{C}_2(50{)}^{49}-........{+}^{51}{C}_{50}(50\left){-}^{51}{C}_{51}\right(1\left)\right]$
$=7{[}^{51}{C}_0\left(50{)}^{51}{-}^{51}{C}_1\right(50{)}^{50}{+}^{51}{C}_2(50{)}^{49}-........{+}^{51}{C}_{50}(50)-1]$
$=7\left(50\right){[}^{51}{C}_0\left(50{)}^{50}{-}^{51}{C}_1\right(50{)}^{49}{+}^{51}{C}_2(50{)}^{48}-........{+}^{51}{C}_{50}(1\left)\right]-7$
$=25k-7$, where $k=7\left(2\right){[}^{51}{C}_0\left(50{)}^{50}{-}^{51}{C}_1\right(50{)}^{49}{+}^{51}{C}_2(50{)}^{48}-........{+}^{51}{C}_{50}(1\left)\right]$
$=25k+(18-25)=25(k-1)+18$
Hence remainder is $18$