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

The correct answer is Option (1) → 127

\(a^m-b^m\) is always divisible by \(a^k-b^k\) when \(k\) divides \(m\), because \(a^m-b^m=(a^k)^{m/k}-(b^k)^{m/k}\) and \(x^j-y^j\) always has the factor \(x-y\). Here the power \(6n\) is a multiple of 3 for every \(n\), so \(k=3\) can be used.

• \(7^{6n}-6^{6n}=(7^3)^{2n}-(6^3)^{2n}\), which is divisible by \(7^3-6^3\)

• \(7^3-6^3=343-216=127\)

The other options fail already for \(n=1\): \(7^6-6^6=117649-46656=70993=127\times559\), an odd number, so it cannot be divided by 556, and it leaves a remainder when divided by 17 or 23.