SATHEE: Binomial Theorem 2 Question 3

Binomial Theorem 2 Question 3

3.

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

(2019 Main, 9 Jan I)

(a) 14

(b) 6

(d) 8

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Answer:

Correct Answer: 3. (d)

Solution:

Consider,

$\begin{aligned} 2^{403} & = 2^{400 + 3} = 8 \cdot 2^{400} = 8 \cdot {(2^{4})}^{100} = 8 (16)^{100} = 8 (1 + 15)^{100} \\ = 8 (1 +^{100} C_{1} (15) +^{100} C_{2} (15)^{2} + \dots +^{100} C_{100} (15)^{100}) \end{aligned}$

[By binomial theorem,

$(1 + x)^{n} =^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x^{2} + \dots^{n} C_{n} x^{n}, n \in N]$

$= 8 + 8 (^{100} C_{1} (15) +^{100} C_{2} (15)^{2} + \dots +^{100} C_{100} (15)^{100})$

$= 8 + 8 \times 15 λ$

where $λ^{100} C_{1} + \dots . . +^{100} C_{100} (15)^{99} \in N$

$∴ \frac{2^{403}}{15} = \frac{8 + 8 \times 15 λ}{15} = 8 λ + \frac{8}{15}$

$\Rightarrow (\frac{2^{403}}{15}) = \frac{8}{15}$

(where { . } is the fractional part function)

$∴ k = 8$

Alternate Method

$2^{403} = 8 \cdot 2^{400} = 8 (16)^{100}$

Note that, when $16$ is divided by $15$ , gives remainder $1$ .

$∴$ When $(16)^{100}$ is divided by $15$ , gives remainder $1^{100} = 1$ and when $8 (16)^{100}$ is divided by $15$ , gives remainder $8$ .

$∴ (\frac{2^{403}}{15}) = \frac{8}{15}$ .

(where { . } is the fractional part function)

$\Rightarrow k = 8$

Binomial Theorem 2 Question 3

3.

Answer:

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Binomial Theorem 2 Question 3

3.

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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