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 3 )^ 25 is an odd number. In the light of the above statements, choose the correct

Correct answer: B. Both Statement I and Statement II are true Statement I: 25^ 13 + 3^ 13 → divisible by (25 + 3) 20^ 13 + 8^ 13 → divisible by (20 + 8) divisible by 7 . TRUE Statement II: For (7+4 3 )^ 25 , let α = 7+4 3 and β = 7-4 3 (conjugate). Then αβ = 1 and α + β = 14 . Since 0 The fractional part of α^ 25 is 1-β^ 25 , making the integral part odd. TRUE. Both Statement I and Statement II are true.

JEE Main 202628 JanuaryEvening Shift

Exam: JEE Main
Year: 2026
Date: 28 January
Shift: Evening Shift
Subject: Mathematics
Chapter: Binomial Theorem

JEE Main 2026 · 28 January · Evening Shift · Mathematics · Binomial Theorem

Given below are two statements : Statement I: 25^ 13 +20^ 13 +8^ 13...

Given below are two statements :
Statement I:

\quad 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

AStatement I is true but Statement II is false
BBoth Statement I and Statement II are true
CStatement I is false but Statement II is true
DBoth Statement I and Statement II are false

Correct answer

B. Both Statement I and Statement II are true

Step-by-step solution

Statement I:

25^{13} + 3^{13}

→ divisible by

(25 + 3)

20^{13} + 8^{13}

→ divisible by

(20 + 8)

\therefore

divisible by

7

. TRUE
Statement II:
For

(7+4\sqrt{3})^{25}

, let

\alpha = 7+4\sqrt{3}

and

\beta = 7-4\sqrt{3}

(conjugate).
Then

\alpha\beta = 1

and

\alpha + \beta = 14

.
Since

0 < \beta < 1

, we have

\alpha^{25} + \beta^{25}

is an integer.
The fractional part of

\alpha^{25}

1-\beta^{25}

, making the integral part odd. TRUE.
Both Statement I and Statement II are true.

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