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

Given: $7^{6n} - 6^{6n}$, where $n$ is a positive integer

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Rewrite the expression:

$7^{6n} - 6^{6n} = (7^6)^n - (6^6)^n$

Let $A = 7^6$ and $B = 6^6$

The expression becomes $A^n - B^n$

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For any positive integer $n$, the expression $A^n - B^n$ is always divisible by $(A - B)$.

Therefore, $(7^6)^n - (6^6)^n$ is always divisible by $(7^6 - 6^6)$.

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Calculate $7^6 - 6^6$:

$7^6 = 117,649$

$6^6 = 46,656$

$7^6 - 6^6 = 117,649 - 46,656 = 70,993$

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Testing divisibility of $70,993$:

$70,993 \div 127 = 559$ (exact division)

$70,993 \div 556 = 127.69...$ (not exact)

$70,993 \div 17 = 4176.06...$ (not exact)

$70,993 \div 23 = 3086.65...$ (not exact)

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Since $(7^6)^n - (6^6)^n$ is always divisible by $(7^6 - 6^6) = 70,993$, and $70,993$ is divisible by $127$, the expression $7^{6n} - 6^{6n}$ is always divisible by $127$ for any positive integer $n$.

Therefore, the answer is $127$.