Question

Prove that n² - 1 is divisible by 12 if g.c.d. (n, 6) = 1

Answer 1

Given gcd(n,6) = 1, it implies that n is coprime with 6, i.e., n and 6 share no common factors other than 1.

Let's consider the expression n² − 1 and try to factorize it:

n² − 1 = (n − 1)(n + 1)

Now, we can analyze this expression modulo 3 and modulo 4:

From the above calculations, we see that n² − 1 is congruent to -1 modulo 3 and modulo 4.

Since n² − 1 is congruent to -1 modulo both 3 and 4, it implies that n² − 1 is congruent to -1 modulo 12.

Thus, n² − 1 is divisible by 12.