Final answer:
To prove that n¹² -1 is divisible by 7 when (n,7) = 1, we can use modular arithmetic and Fermat's Little Theorem. By factoring n¹² - 1 and applying the theorem, we can show that the expression is congruent to 2 modulo 7.
Step-by-step explanation:
To prove that n¹² -1 is divisible by 7 when (n,7) = 1, we can use modular arithmetic. First, let's express n¹² -1 as (n⁶)² - 1. Using the difference of squares, this can be factored as (n⁶ + 1)(n⁶ - 1). Now, we can apply Fermat's Little Theorem which states that if p is a prime number and a is an integer not divisible by p, then a^(p-1) ≡ 1 (mod p). In this case, p = 7, and since (n,7) = 1, we can deduce that n⁶ ≡ 1 (mod 7). Therefore, (n⁶ + 1)(n⁶ - 1) ≡ 1 + 1 ≡ 2 (mod 7), which implies that n¹² - 1 is divisible by 7.