Given below are two statements : Statement I: 25¹³ + 20¹³ + 8¹³ + 3¹³ is divisible by 7. Statement I

Q: Wait, why can't we just calculate the values of $25^{13}$ and the others? It's just arithmetic, right?

I love the enthusiasm, but in JEE Advanced, we are testing your ability to see patterns, not your ability to do massive arithmetic! Calculating those powers would take hours and lead to errors. The identity $a^n + b^n = (a+b)(a^{n-1} - a^{n-2}b + \dots + b^{n-1})$ is the elegant shortcut designed to save you time and ensure accuracy.

Q: Why does the sum of the fractional parts $f + f'$ have to be exactly $1$? Couldn't it be something else?

That is a brilliant question. We know $0 < f < 1$ and $0 < f' < 1$. If we add these inequalities, we get $0 < f + f' < 2$. Since $I + f + f'$ must be an integer (because the binomial expansion sum is an integer), and $I$ is an integer, $f + f'$ must also be an integer. The only integer strictly between $0$ and $2$ is $1$. There is no other possibility!

Q: How are we so sure that $(7 + 4\sqrt{3})^{25} + (7 - 4\sqrt{3})^{25}$ is an even integer?

Think about the Binomial Theorem. When you expand $(a+b)^n + (a-b)^n$, the terms with odd powers of $b$ have opposite signs and cancel out perfectly. The remaining terms all have even powers of $b$. Since $b = 4\sqrt{3}$, $b^2 = 48$, which is an integer. Thus, all remaining terms are integers, and because we have two of each, the sum is guaranteed to be even.

Question

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 :

Options:

(A) Statement I is true but Statement II is false

(B) Both Statement I and Statement II are true

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

(D) Both Statement I and Statement II are false

Sigma Percentile · Accepted Answer

### Analyzing the Divisibility Pattern

Consider the expression $25^{13} + 20^{13} + 8^{13} + 3^{13}$. When dealing with large powers in JEE problems, we look for algebraic identities.

Recall that for any odd natural number $n$, the expression $a^n + b^n$ is always divisible by $(a+b)$. We group the terms strategically to exploit this property:
$$(25^{13} + 3^{13}) + (20^{13} + 8^{13})$$

Since $25 + 3 = 28$ and $20 + 8 = 28$, both pairs are divisible by $28$. Because $28$ is a multiple of $7$, both $(25^{13} + 3^{13})$ and $(20^{13} + 8^{13})$ are divisible by $7$.

Consequently, their sum is also a multiple of $7$. Therefore, **Statement I is true**.

### The Conjugate Method for Integral Parts

Let the expression be $(7 + 4\sqrt{3})^{25} = I + f$, where $I$ is the integer part and $0 < f < 1$. We define the conjugate term $f' = (7 - 4\sqrt{3})^{25}$.

Since $4\sqrt{3} = \sqrt{48} \approx 6.928$, it follows that $0 < 7 - 4\sqrt{3} < 1$. Raising this value to the power of $25$ ensures that $0 < f' < 1$.

Now, consider the sum:
$$(I + f) + f' = (7 + 4\sqrt{3})^{25} + (7 - 4\sqrt{3})^{25}$$

### Evaluating the Integer Part

Applying the Binomial Theorem, the odd powers of $4\sqrt{3}$ cancel out. This leaves us with:
$$2 \sum_{k=0, 2, 4...}^{24} \binom{25}{k} 7^{25-k} (4\sqrt{3})^k$$

Since all terms in this expansion are integers, the sum $(I + f) + f'$ is an **even integer**. Because $I$ is an integer, $f + f'$ must also be an integer.

Given the constraints $0 < f < 1$ and $0 < f' < 1$, the sum $f + f'$ must satisfy $0 < f + f' < 2$. The only integer in this range is $1$, so $f + f' = 1$.

Substituting this into our equation:
$$I + 1 = 	ext{Even Integer}$$
$$I = 	ext{Even Integer} - 1$$

An even number minus one is always odd. Thus, the integral part $I$ is odd, and **Statement II is true**.

Select Answer:

Visualized Solution

$Analyzing the Statements$

$Divisibility Rule for a^{n} + b^{n}$

$Grouping the Terms$

$Analyzing the First Pair$

$Analyzing the Second Pair$

$Conclusion for Statement I$

$Setting up Statement II$

$Defining the Conjugate Term$

$Sum of the Two Expansions$

$Analyzing the Fractional Sum$

$Determining the Parity of I$

$Final Conclusion$

The Sigma Insight: Binomial Expansion for Positive Integral Index

Analyzing the Divisibility Pattern

The Conjugate Method for Integral Parts

Evaluating the Integer Part

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Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Analyzing the Statements

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Divisibility Rule for an+bn

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Grouping the Terms

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Analyzing the First Pair

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Analyzing the Second Pair

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Conclusion for Statement I

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Setting up Statement II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Defining the Conjugate Term

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Sum of the Two Expansions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Analyzing the Fractional Sum

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Determining the Parity of I

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Final Conclusion