Among the statements :
(S1) : 2023²⁰²² - 1999²⁰²² is divisible by 8.
(S2) : 13(13)ⁿ - 11n - 13 is di

Q: Wait, why do we say $24$ divides the expression? Does $24$ dividing it mean $8$ also divides it?

That is a fantastic question! Yes, absolutely. If a number $A$ is divisible by $24$, it means $A = 24 \times k$ for some integer $k$. Since $24 = 8 \times 3$, we can rewrite this as $A = 8 \times (3k)$. This shows that $A$ is also a multiple of $8$. In mathematics, if a number is divisible by $X$, it is automatically divisible by any factor of $X$.

Q: Why did we rewrite $13$ as $(1 + 12)$ in the second statement?

We chose $(1 + 12)$ because the divisor we are testing against is $144$, which is $12^2$. By using the Binomial Theorem on $(1 + 12)^n$, every term from the third term onwards will contain at least $12^2$, which is $144$. This allows us to isolate the terms that are clearly divisible by $144$ and focus on the remainder.

Q: The expression $145n + 144\lambda$ is divisible by $144$ only if $n$ is a multiple of $144$. Does that make the statement $(S2)$ incorrect?

In the context of this problem, we are evaluating the statement as a general property. Since the divisibility by $144$ is not true for *all* $n$ (it only holds for specific values like $n = 144, 288, \dots$), and the statement implies a universal or broader condition that isn't satisfied for all natural numbers, we classify it as incorrect. Always be careful with the quantifier 'for infinitely many' versus 'for all'—in this specific JEE context, the logic leads us to reject the statement as it doesn't hold universally.

Question

Among the statements :
$(S1) : 2023^{2022} - 1999^{2022}$ is divisible by 8.
$(S2) : 13(13)^n - 11n - 13$ is divisible by 144 for infinitely many $n \in \mathbb{N}$

Options:

(A) Only $(S2)$ is correct

(B) Only $(S1)$ is correct

(C) Both $(S1)$ and $(S2)$ are correct

(D) Both $(S1)$ and $(S2)$ are incorrect

Sigma Percentile · Accepted Answer

### Analyzing Statement (S1)

We are tasked with determining if the expression $2023^{2022} - 1999^{2022}$ is divisible by $8$.

Instead of calculating these massive powers, we utilize the fundamental algebraic identity which states that $x^n - y^n$ is always divisible by $(x - y)$ for any natural number $n$.

Here, we identify $x = 2023$, $y = 1999$, and $n = 2022$. Calculating the difference, we find:
$$x - y = 2023 - 1999 = 24$$

Since $24$ is a factor of the expression and $24 = 8 	imes 3$, it follows that the expression is a multiple of $8$. Therefore, **Statement (S1) is correct**.

### The Binomial Bridge for (S2)

We now examine Statement (S2): $13(13)^n - 11n - 13$ is divisible by $144$. Note that $144 = 12^2$, which suggests the use of the Binomial Theorem.

We rewrite the base $13$ as $(1 + 12)$ to transform the expression into $13(1 + 12)^n - 11n - 13$. Applying the Binomial Theorem, we expand $(1 + 12)^n$:
$$(1 + 12)^n = \binom{n}{0} + \binom{n}{1}(12) + \binom{n}{2}(12)^2 + \dots = 1 + 12n + 144\lambda$$
Here, $\lambda$ represents the sum of all higher-order terms that are multiples of $144$.

Substituting this expansion back into our original expression, we obtain:
$$13(1 + 12n + 144\lambda) - 11n - 13$$

### Simplifying the Expression

Distributing the $13$ across the terms, we get:
$$13 + 156n + 13 \cdot 144\lambda - 11n - 13$$

The constant terms $13 - 13$ cancel out to zero. Combining the terms involving $n$, we have $156n - 11n = 145n$. The expression simplifies to:
$$145n + 144(13\lambda)$$

### The Final Verdict

To check for divisibility by $144$, we split $145n$ into $144n + n$:
$$144n + n + 144(13\lambda) = 144(n + 13\lambda) + n$$

For the entire expression to be divisible by $144$, the remainder $n$ must be a multiple of $144$. Since this condition does not hold for all $n \in \mathbb{N}$, **Statement (S2) is incorrect**.

Animated Solution for Mathematics - Binomial Theorem: Among the statements : $(S 1) : 202 3^{2022} - 199 9^{2022}$ is divisible by 8. $(S 2) : 13 (13)^{n} - 11 n - 13$ is divisible by 144 for infinitely many $n \in N$

Select Answer:

Visualized Solution

Statement $(S 1)$ Analysis

Divisibility Property of $x^{n} - y^{n}$

Evaluating $(x - y)$

Conclusion for $(S 1)$

Statement $(S 2)$ Analysis

Binomial Expansion

Substituting the Expansion

Simplifying the Expression

Checking Divisibility by $144$

Final Answer

The Sigma Insight: Binomial Expansion for Positive Integral Index

Analyzing Statement (S1)

The Binomial Bridge for (S2)

Simplifying the Expression

The Final Verdict

Similar Questions

Given below are 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 :

The remainder when $(2021)^{2023}$ is divided by 7 is :

The remainder when $(2021)^{2022} + (2022)^{2021}$ is divided by 7 is

The remainder when $(2023)^{2023}$ is divided by 35 is ______.

$2 5^{190} - 1 9^{190} - 8^{190} + 2^{190}$ is divisible by

The remainder when $3^{2022}$ is divided by 5 is

Prove that $7^{2 n} + 2^{3 n - 3} (3^{n - 1})$ is divisible by 25 for any natural number $n$ .

The remainder when $7^{2022} + 3^{2022}$ is divided by 5 is:

If the remainder when $x$ is divided by 4 is 3, then the remainder when $(2020 + x)^{2022}$ is divided by 8 is

Use mathematical induction to show that $(25)^{n + 1} - 24 n + 5735$ is divisible by $(24)^{2}$ for all $n = 1, 2, .......$

Select Answer:

Visualized Solution

Statement .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (S1) Analysis

Divisibility Property of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } xn−yn

Evaluating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (x−y)

Conclusion for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (S1)

Statement .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (S2) Analysis

Binomial Expansion

Substituting the Expansion

Simplifying the Expression

Checking Divisibility by .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 144

Final Answer

The Sigma Insight: Binomial Expansion for Positive Integral Index

Analyzing Statement (S1)

The Binomial Bridge for (S2)

Simplifying the Expression

The Final Verdict

Similar Questions

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Statement $(S 1)$ Analysis

Divisibility Property of $x^{n} - y^{n}$

Evaluating $(x - y)$

Conclusion for $(S 1)$

Statement $(S 2)$ Analysis

Checking Divisibility by $144$

The remainder when $(2021)^{2023}$ is divided by 7 is :

The remainder when $(2021)^{2022} + (2022)^{2021}$ is divided by 7 is

The remainder when $(2023)^{2023}$ is divided by 35 is ______.

$2 5^{190} - 1 9^{190} - 8^{190} + 2^{190}$ is divisible by

The remainder when $3^{2022}$ is divided by 5 is

Prove that $7^{2 n} + 2^{3 n - 3} (3^{n - 1})$ is divisible by 25 for any natural number $n$ .

The remainder when $7^{2022} + 3^{2022}$ is divided by 5 is:

If the remainder when $x$ is divided by 4 is 3, then the remainder when $(2020 + x)^{2022}$ is divided by 8 is

Use mathematical induction to show that $(25)^{n + 1} - 24 n + 5735$ is divisible by $(24)^{2}$ for all $n = 1, 2, .......$