Q. Given below are two statements :

Statement I :
25¹³ + 20¹³ + 8¹³ + 3¹³ 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 answer from the options given below :

Correct Answer: Both Statement I and Statement II are true

Explanation

Statement I:

Consider each term modulo 7.

25 ≡ 4 (mod 7), 20 ≡ 6 (mod 7), 8 ≡ 1 (mod 7), 3 ≡ 3 (mod 7)

Now,

4¹³ ≡ 4 (mod 7)
6¹³ ≡ 6 (mod 7)
1¹³ ≡ 1 (mod 7)
3¹³ ≡ 3 (mod 7)

Adding,

4 + 6 + 1 + 3 = 14 ≡ 0 (mod 7)

Hence, the given expression is divisible by 7.

So, Statement I is true.

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Statement II:

Note that

(7 + 4√3)(7 − 4√3) = 49 − 48 = 1

So,

(7 − 4√3) = 1 / (7 + 4√3)

Since 0 < 7 − 4√3 < 1, we have

(7 − 4√3)²⁵ < 1

Now consider

(7 + 4√3)²⁵ + (7 − 4√3)²⁵

This sum is an integer and also an odd number.

Hence,

(7 + 4√3)²⁵ = odd integer − (7 − 4√3)²⁵

Since (7 − 4√3)²⁵ < 1, the integral part of (7 + 4√3)²⁵ is that odd integer.

So, Statement II is true.

Therefore, the correct answer is:

Both Statement I and Statement II are true

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