For a positive integer n, 3⁶ⁿ − 6³ⁿ is divisible by:

Solution
Given:
3⁶ⁿ − 6³ⁿ
First simplify:
3⁶ⁿ = (3²)³ⁿ
= 9³ⁿ
So,
3⁶ⁿ − 6³ⁿ
= 9³ⁿ − 6³ⁿ
Using identity:
a³ − b³ = (a − b)(a² + ab + b²)
Here,
a = 9ⁿ
b = 6ⁿ
Therefore,
9³ⁿ − 6³ⁿ
= (9ⁿ − 6ⁿ)(9²ⁿ + 9ⁿ6ⁿ + 6²ⁿ)
Now,
9ⁿ − 6ⁿ
= 3ⁿ(3ⁿ − 2ⁿ)
Hence the whole expression contains a factor 3ⁿ.
Also,
9³ⁿ − 6³ⁿ
= (3²)³ⁿ − (2 × 3)³ⁿ
= 3⁶ⁿ − 2³ⁿ × 3³ⁿ
= 3³ⁿ(3³ⁿ − 2³ⁿ)
Thus the expression is always divisible by:
3³ⁿ
Final Answer
The expression is divisible by 3³ⁿ.