Question

Find the remainder when 7${}^{103}$ is divided by $25.$

• A. $7$
• B. $24$
• C. $18$
• D. $1$

Answer 1

Answer: (C)

$18$

$7^{103}$
$=7^{102}\left(7\right)$
$=7(49{)}^{51}$
$=7(50-1{)}^{51}$
$=7({50}^{51}-51(50{)}^{50}+\left(51\right)\left(25\right)(50{)}^{49}...+(51\left)\right(50)-1)$
$=7(50k-1)$
$=350k-7$
$=350k+25-25-7$
$=(350k-25)+25-7$
$=(350k-25)+18$ ...(a)
When '$a$' is divided by $25$ it leaves a remainder $18$.
Hence, when $7^{103}$ is divided by $25$, it leaves a remainder $18$.