Divisibility of 9²ⁿ − 4²ⁿ for Natural Number n

Video Explanation

Watch the video below for a clear explanation:

Solution

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

(a) 5    (b) 13    (c) both 5 and 13    (d) None of these

Step 1: Rewrite the Expression

9²ⁿ = (3²)²ⁿ = 3⁴ⁿ

4²ⁿ = (2²)²ⁿ = 2⁴ⁿ

∴ 9²ⁿ − 4²ⁿ = 3⁴ⁿ − 2⁴ⁿ

Step 2: Use Identity

aⁿ − bⁿ is divisible by (a − b)

Here, a = 3⁴ and b = 2⁴

So,

3⁴ⁿ − 2⁴ⁿ is divisible by (3⁴ − 2⁴)

= 81 − 16

= 65

Step 3: Factorise 65

65 = 5 × 13

∴ The given expression is divisible by both 5 and 13.

Final Answer

✔ 9²ⁿ − 4²ⁿ is always divisible by both 5 and 13.

✔ Correct option: (c) both 5 and 13

Conclusion

Thus, for every natural number n, the expression 9²ⁿ − 4²ⁿ is divisible by both 5 and 13.