Remainder of $(54)^{53}$ when divided by $11$ is

1

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7

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9

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10

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Solution

The correct option is D $10$
$(54)^{53}$ divided by 11
$(54)^{53} = (55 - 1)^{53}$
$= 55^{33} - (55)^{52} .^{53} C_{1} +^{53} C_{2} (55)^{51} - . . .$
$. . . -^{53} C_{51} 55^{2} +^{53} C_{52} 55 - 1$
$= 55 K - 1$
$= 55 (K - 1) + 55 - 1$
$= 55 K^{'} + 54$
$= 55 K^{'} + 44 + 10$
$= 11 (5 K^{'} + 4) + 10$
$= 11 K_{1} + 10$
$\Rightarrow$ Remainder = 10

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Remainder of (54)53 when divided by 11 is

The correct option is D 10(54)53 divided by 11 (54)53=(55−1)53=5533−(55)52.53C1+53C2(55)51−... ...−53C51552+53C5255−1=55K−1=55(K−1)+55−1=55K′+54=55K′+44+10=11(5K′+4)+10=11K1+10⇒ Remainder = 10

Remainder of $(54)^{53}$ when divided by $11$ is