The remainder, when 7 103 is divided by 17, is ________ . [2023]

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Solution

Q.
The remainder, when $7^{103}$ is divided by 17, is ________ . [2023]

Ans.
(12)

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

$7 . 7^{102} = 7 {(7^{2})}^{51} = 7 {(49)}^{51} = 7 {(51 - 2)}^{51}$

$= 7 {(- 2)}^{51} = - 7 {(2)}^{51} = - 7 (2^{48 + 3}) = - 7.8 (2^{48})$

$= - 56 {(2^{4})}^{12} = - 56 {(17 - 1)}^{12} = - 56 \times {(- 1)}^{12} = (- 56)$

$Remainder = 12$

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