If n is a natural number, then 9²ⁿ 4²ⁿ is always divisible by

Given:

If n is a natural number, then 92ⁿ - 42ⁿ is always divisible by:

Options:

1) 5

2) 13

3) Both (A) and (B)

4) Neither (A) nor (B)

Formula used:

For divisibility rules, we check if the expression satisfies modulo conditions for the given numbers.

92 ≡ -3 (mod 5), 42 ≡ 2 (mod 5)

92 ≡ 1 (mod 13), 42 ≡ 3 (mod 13)

Calculations:

Step 1: Modulo 5 check

92ⁿ ≡ (-3)ⁿ (mod 5), 42ⁿ ≡ 2ⁿ (mod 5)

⇒ 92ⁿ - 42ⁿ ≡ (-3)ⁿ - 2ⁿ (mod 5)

For all n, (-3)ⁿ ≡ 2ⁿ (mod 5), hence divisible by 5.

Step 2: Modulo 13 check

92ⁿ ≡ 1ⁿ (mod 13), 42ⁿ ≡ 3ⁿ (mod 13)

⇒ 92ⁿ - 42ⁿ ≡ 1ⁿ - 3ⁿ (mod 13)

For all n, 1ⁿ - 3ⁿ ≡ 0 (mod 13), hence divisible by 13.

Step 3: Combine results

Since 92ⁿ - 42ⁿ is divisible by both 5 and 13, it is divisible by both.

∴ The correct answer is option (3).