Question:
If the fractional part of the number $\frac{2^{403}}{15}$ is $\frac{k}{15}$, then $k$ is equal to :
If the fractional part of the number $\frac{2^{403}}{15}$ is $\frac{k}{15}$, then $k$ is equal to :
Updated On: May 23, 2025
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The Correct Option is D
Solution and Explanation
$\frac{2^{403}}{15} = \frac{2^{3}. \left(2^{4}\right)^{100}}{15} = \frac{8}{15} \left(15+1\right)^{100} $
$ = \frac{8}{15} \left(15 \lambda +1\right) = 8 \lambda + \frac{8}{15} $
$ \because 8 \lambda$ is integer
$ \Rightarrow $ fractional part of $ \frac{2^{403}}{15} \frac{8}{15} \Rightarrow k = 8 $
$ = \frac{8}{15} \left(15 \lambda +1\right) = 8 \lambda + \frac{8}{15} $
$ \because 8 \lambda$ is integer
$ \Rightarrow $ fractional part of $ \frac{2^{403}}{15} \frac{8}{15} \Rightarrow k = 8 $
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.