Given below are two statements:Statement I: 2 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Statement $I$: We have $25^{13} + 3^{13} + 20^{13} + 8^{13}$.Since $a^n + b^n$ is divisible by $(a + b)$ when $n$ is odd, we have:$25^{13} + 3^{13

Vedclass · Accepted Answer

Both Statement $I$ and Statement $II$ are true.

Vedclass · Answer

(B) Statement $I$: We have $25^{13} + 3^{13} + 20^{13} + 8^{13}$.Since $a^n + b^n$ is divisible by $(a + b)$ when $n$ is odd, we have:$25^{13} + 3^{13}$ is divisible by $(25 + 3) = 28$, which is divisible by $7$.$20^{13} + 8^{13}$ is divisible by $(20 + 8) = 28$, which is divisible by $7$.Thus, the sum is divisible by $7$. Statement $I$ is true.Statement $II$: Let $R = (7 + 4\sqrt{3})^{25} = I + f$, where $I$ is the integral part and $0 Let $R' = (7 - 4\sqrt{3})^{25} = f'$, where $0 Since $7^2 - (4\sqrt{3})^2 = 49 - 48 = 1$, $R'$ is a very small positive number.$R + R' = (7 + 4\sqrt{3})^{25} + (7 - 4\sqrt{3})^{25} = 2 \left[ {}^{25}C_0 7^{25} + {}^{25}C_2 7^{23}(4\sqrt{3})^2 + \dots ight]$.This is an even integer. So, $I + f + f' = 	ext{even integer}$.Since $0 Therefore, $I + 1 = 	ext{even integer}$, which implies $I$ is an odd integer. Statement $II$ is true.

Given below are two statements:
Statement $I$: $25^{13} + 20^{13} + 8^{13} + 3^{13}$ is divisible by $7$.
Statement $II$: The integral part of $(7 + 4\sqrt{3})^{25}$ is an odd number.
In the light of the above statements, choose the correct answer from the options given below:

Similar Questions

Let $n \in \mathbb{N}$. Which one of the following is true?

The remainder when $3^{2022}$ is divided by $5$ is

$7^{2n} + 16n - 1$ $(n \in N)$ is divisible by

The number $(101)^{100}-1$ is divisible by

Let $a, b, c, d$ be positive integers. Consider the following statements:
$I$. If $9$ divides $a^3+b^3+c^3$,then $3$ divides $abc$.
$II$. If $9$ divides $a^3+b^3+c^3+d^3$,then $3$ divides $abcd$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module

Given below are two statements:Statement $I$: $25^{13} + 20^{13} + 8^{13} + 3^{13}$ is divisible by $7$.Statement $II$: The integral part of $(7 + 4\sqrt{3})^{25}$ is an odd number.In the light of the above statements, choose the correct answer from the options given below: