7⁶ⁿ - 6⁶ⁿ, where n is an integer > 0, is divisible by :

When n is even, (xⁿ - aⁿ) is divisible by both (x - a) as well as (x + a).
Now,
(7⁶ⁿ - 6⁶ⁿ)
= [(7³)²ⁿ - (6³)²ⁿ]
= [(343)²ⁿ - (216)²ⁿ]
∴ (7⁶ⁿ - 6⁶ⁿ) is divisible by both (7 - 6) and (7 + 6)
(7⁶ⁿ - 6⁶ⁿ) is divisible by 13
And, [(343)²ⁿ - (216)²ⁿ] is divisible by both (343 - 216) and (343 + 216)
⇒ (7⁶ⁿ - 6⁶ⁿ) is divisible by both 127 and 559