Fractional part of the number 4 2022 15 is equal to

Given that 4 2022 15 We know that a + x n = C 0 n a n x 0 + C 1 n a n - 1 x 1 + . . . . . + C n n a 0 x n ⇒ 4 2022 15 = 16 1011 15 = 15 + 1 1011 15 = C 0 1011 15 1011 + C 1 1011 15 1010 + . . . . . + C 1011 1011 15 0 15 = C 0 1011 15 1011 + C 1 1011 15 1010 + . . . . . + C 1010 1011 15 1 15 + 1 15 ∴ Fractional part of 4 2022 15 = 1 15 Hence this is the required option.

Mathematics - Binomial Theorem Question with Solution | TestHub

MathematicsBinomial TheoremRemainder and Divisibility ProblemsEasy2 minPYQ_2023

MathematicsEasysingle choice

Fractional part of the number $\frac{4^{2022}}{15}$ is equal to

Options:

Answer:

Solution:

Given that $\frac{4^{2022}}{15}$
We know that ${(a + x)}^{n} = {}^{n}C_{0} a^{n} x^{0} + {}^{n}C_{1} a^{n - 1} x^{1} + . . . . . + {}^{n}C_{n} a^{0} x^{n}$
$\Rightarrow \frac{4^{2022}}{15} = \frac{16^{1011}}{15} = \frac{{(15 + 1)}^{1011}}{15}$
$= \frac{{}^{1011}C_{0} {(15)}^{1011} + {}^{1011}C_{1} {(15)}^{1010} + . . . . . + {}^{1011}C_{1011} {(15)}^{0}}{15}$
$= \frac{{}^{1011}C_{0} {(15)}^{1011} + {}^{1011}C_{1} {(15)}^{1010} + . . . . . + {}^{1011}C_{1010} {(15)}^{1}}{15} + \frac{1}{15}$
$∴$ Fractional part of $\frac{4^{2022}}{15} = \frac{1}{15}$
Hence this is the required option.

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

⏱ 2mℹ️ Source: PYQ_2023

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