Binomial TheoremHard

Question

Given below 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:

Options

A.Both Statement I and Statement II are false.
B.Both Statement I and Statement II are true.
C.Statement I is false but Statement II is true.
D.Statement I is true but Statement II is false.

Solution

Statement I :

Statement II : $R = (7 + 4\sqrt{3})^{25} = I + f$

$${R' = (7 - 4\sqrt{3})^{25} = f' }{\therefore R + R' = 2\left\lbrack \ ^{25}C_{0}7^{25} + \ ^{25}C_{2}7^{23}(4\sqrt{3})^{2} + \ldots. \right\rbrack }$$$I + f + f' =$ even integer

$\therefore I =$ odd integer

$$\because 0 < f + f' < 2 \Rightarrow f + f' = 1 $$⇒ Both the statements are correct

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