Question

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

• A. $20$
• B. $16$
• C. $18$
• D. $15$

Answer 1

The correct option is C $18$

$7^{103}$ mod $25=?$

$\Rightarrow 7\times 7^{102}$

$\Rightarrow 7\times (49{)}^{51}$

$\Rightarrow 7\times (50-1{)}^{51}$

$\Rightarrow -7\times (1-50{)}^{51}$

$\Rightarrow -7\times [1-{}^{51}{C}_1(50)+{}^{51}{C}_2(50{)}^2+.....]$

$\Rightarrow -7+50k$

$\Rightarrow -7+25\left(p\right)$

$\Rightarrow -7+25+25(p-1)$

$\Rightarrow 18+25\left(M\right)$

$\therefore 7^{103}=18+25\left(M\right)$

$\therefore$ Remainder when $7^{103}$ is divided by $25=18$.