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 :

(B) Both Statement I and Statement II are true.

(A) Both Statement I and Statement II are false.

(C) Statement I is false but Statement II is true.

(D) Statement I is true but Statement II is false.

Binomial Theorem — JEE Main · 2026 · 28 Jan 2026 · Shift 2 Question with Solution

Given below two statements :

Statement I :

2 5^{13} + 2 0^{13} + 8^{13} + 3^{13}

is divisible by 7.
Statement II : The integral part of

(7 + 43)^{25}

is an odd number.

In the light of the above statements, choose the correct answer from the options given below :

Answer: (B)

For Statement I, group the terms:

2 5^{13} + 3^{13}

is divisible by

25 + 3 = 28

and

2 0^{13} + 8^{13}

is divisible by

20 + 8 = 28

, since

a^{n} + b^{n}

is divisible by

a + b

for odd

n

. As

28 = 4 \cdot 7

, the whole sum is divisible by

7

, so Statement I is true.

For Statement II, since

0 < 7 - 43 < 1

, write

(7 + 43)^{25} = I + f

with

0 < f < 1

, and let

f^{'} = (7 - 43)^{25} \in (0, 1)

. In

(7 + 43)^{25} + (7 - 43)^{25}

the irrational parts cancel, leaving an even integer. Thus

I + f + f^{'}

is an even integer, and since

0 < f + f^{'} < 2

it follows that

f + f^{'} = 1

. Then

I + 1

is even, so

I

is odd, and Statement II is true.

Both Statement I and Statement II are true.

JEE Main · 2026 · 28 Jan 2026 · Shift 2 · Single correct option