$7^{6n} - 6^{6n}$, where $n$ is an integer greater than 0, is divisible by

127

The correct answer is Option (1) → 127

1. Algebraic Identity

The expression $a^k - b^k$ is always divisible by $(a - b)$ for any positive integer $k$.

In the given expression, $7^{6n} - 6^{6n}$, we can rewrite it as:

$(7^6)^n - (6^6)^n$

Thus, for any $n > 0$, the expression is divisible by $(7^6 - 6^6)$.

2. Factoring the term

We can further factorize $7^6 - 6^6$ using the difference of squares:

$7^6 - 6^6 = (7^3)^2 - (6^3)^2 = (7^3 - 6^3)(7^3 + 6^3)$

Now, let's calculate the values:

• $7^3 = 7 \times 7 \times 7 = 343$
• $6^3 = 6 \times 6 \times 6 = 216$

Substituting these values into the first factor:

$7^3 - 6^3 = 343 - 216 = 127$

3. Conclusion

Since $127$ is a factor of $(7^6 - 6^6)$, and $(7^6 - 6^6)$ is a factor of $7^{6n} - 6^{6n}$, it follows that $7^{6n} - 6^{6n}$ is always divisible by 127.