The remainder on dividing 5 99 by 11 is _________ . [2023]

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Solution

Q.
The remainder on dividing $5^{99}$ by 11 is _________ . [2023]

Ans.
(9)

$We have, 5^{99} = 5^{4} \times 5^{95} = 625 \times {(5^{5})}^{19}$

$= 625 \times {(3125)}^{19} = 625 \times {(3124 + 1)}^{19}$

$= 625 \times {(11 λ + 1)}^{19} [∵ 11 \times 284 = 3124]$

$∴$ $Remainder = 9$

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