Question

Consider the following Assertion (A): \[ 5^{2n}-3^{2n-1} \] is divisible by 2 for all \(n\in\mathbb N\). Reason (R): \[ 5^{2n}+3^{2n-1} \] is divisible by 7 for all \(n\in\mathbb N\).

• A. Both A and R are correct and R is the correct explanation of A
• B. Both A and R are correct and R is not the correct explanation of A
• C. A is correct, but R is not correct
• D. A is not correct, but R is correct \bigskip

Hint:

For Assertion--Reason questions involving divisibility, always test the reason using the smallest possible values such as \(n=1\) or \(n=2\). A single counterexample is enough to disprove a universal statement.

Answer 1

The Correct Option is C

Concept: In divisibility questions, parity (odd/even nature) and modular arithmetic are the most useful tools. We shall examine the assertion and reason independently.

Step 1: Verify Assertion (A).
Observe that \[ 5^{2n} \] is an odd number because any power of an odd number remains odd. Similarly, \[ 3^{2n-1} \] is also odd. Therefore, \[ 5^{2n}-3^{2n-1} \] is the difference of two odd numbers. We know that \[ \text{odd}-\text{odd}=\text{even}. \] Hence \[ 2\mid\left(5^{2n}-3^{2n-1}\right). \] Thus Assertion (A) is true.

Step 2: Verify Reason (R).
Take \[ n=1. \] Then \[ 5^{2}+3^{1} = 25+3 = 28. \] Since \[ 28=7\times4, \] it is divisible by 7. Now take \[ n=2. \] Then \[ 5^{4}+3^{3} = 625+27 = 652. \] Dividing by 7, \[ 652=7(93)+1. \] Thus 652 is not divisible by 7. Therefore the statement is false.

Step 3: Draw the conclusion.
Assertion (A) is true. Reason (R) is false. Hence the correct choice is \[ \boxed{\text{A is correct, but R is not correct}.} \] Therefore, \[ \boxed{(C)} \] is the correct answer.