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JEE Main 202313 AprilMorning Shift
Exam
JEE Main
Year
2023
Date
13 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Binomial Theorem

JEE Main 2023 · 13 April · Morning Shift · Mathematics · Binomial Theorem

Fractional part of the number 4 ^ 2022 15 is equal to

Fractional part of the number 4202215\dfrac{{4}^{2022}}{15} is equal to

Options

  1. A815\dfrac{8}{15}
  2. B415\dfrac{4}{15}
  3. C1415\dfrac{14}{15}
  4. D115\dfrac{1}{15}

Correct answer

D. 115\dfrac{1}{15}

Step-by-step solution

Given that 4202215\dfrac{{4}^{2022}}{15}

We know that (a+x)n=C0nanx0+C1nan−1x1+.....+Cnna0xn{\left(a + x\right)}^{n} = C0 n {a}^{n}{x}^{0} + C1 n {a}^{n - 1}{x}^{1} + . . . . . + Cn n {a}^{0}{x}^{n}

⇒4202215=16101115=(15+1)101115⇒ \dfrac{{4}^{2022}}{15} = \dfrac{{16}^{1011}}{15} = \dfrac{{\left(15 + 1\right)}^{1011}}{15}

=C01011(15)1011+C11011(15)1010+.....+C10111011(15)015= \dfrac{C0 1011 {\left(15\right)}^{1011} + C1 1011 {\left(15\right)}^{1010} + . . . . . + C1011 1011 {\left(15\right)}^{0}}{15}

=C01011(15)1011+C11011(15)1010+.....+C10101011(15)115+115= \dfrac{C0 1011 {\left(15\right)}^{1011} + C1 1011 {\left(15\right)}^{1010} + . . . . . + C1010 1011 {\left(15\right)}^{1}}{15} + \dfrac{1}{15}

∴∴ Fractional part of4202215=115\dfrac{{4}^{2022}}{15} = \dfrac{1}{15}

Hence this is the required option.

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