The remainder, when 7^103 is divided by 17, is __________ Answer: 12

Question

The remainder, when $7^{103}$ is divided by 17, is \_\_\_\_\_\_\_\_\_\_

Answer: 12

Accepted Answer

${7}^{103}=7	imes {7}^{102}$

$\begin{gathered} =7	imes (49{)}^{51} \ =7	imes (51-2{)}^{51} \end{gathered}$

Remainder = $7	imes (-2{)}^{51}$

$\begin{gathered} =-7({2}^{3}\cdot (16{)}^{12}) \ =-56(17-1{)}^{12} \end{gathered}$

Remainder $=-56	imes (-1{)}^{12}=-56+68=12$

Solutions · 1