If the fractional part of the number 2 403 15 is k 15 , then k is equal to

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MathematicsBinomial TheoremRemainder and Divisibility ProblemsMedium2 minPYQ_2019

MathematicsMediumsingle choice

If the fractional part of the number $\frac{2^{403}}{15}$ is $\frac{k}{15},$ then $k$ is equal to

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Answer:

Solution:

$2^{403} = 2^{3} . {(2^{4})}^{100}$
$= 8 . {(15 + 1)}^{100}$
$= 8 [^{100} C_{0} 15^{100} +^{100} C_{1} 15^{99} + . . . . +^{100} C_{99} 15 +^{100} C_{100}]$
$= 8 . [15 λ + 1]$
$= 8.15 λ + 8$
$\Rightarrow \frac{2^{403}}{15} = 8 λ + \frac{8}{15}$
Hence, $k = 8$

Stream:JEESubject:MathematicsTopic:Binomial TheoremSubtopic:Remainder and Divisibility Problems

⏱ 2mℹ️ Source: PYQ_2019

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