Question

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

Answer 1

Since $7^{103}=7(49{)}^{51}=7(50-1{)}^{51}$
$=7\{{50}^{51}{-}^{51}{C}_1{50}^{50}{+}^{51}{C}_2{50}^{49}-\cdots -1\}=7({50}^{51}{-}^{51}{C}_1{50}^{50}{+}^{51}{C}_2{50}^{49}-\cdots )-7+18-18=7({50}^{51}{-}^{51}{C}_1{50}^{50}{+}^{51}{C}_2{50}^{49}-\cdots )-25+18$
= k + 18 (say)
$\because$ k is divisible by 25.
Hence remainder is 18.