# Congruence with Incongruent Solutions

Let n be an integer greater than 2. For which values of n (if any) does the congruence 12x ≡ 8 (mod n) have exactly two incongruent solutions (mod n)? Justify your answer.

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#### Solution Preview

We need to consider the congruence classes of n modulo 12.

First, consider the cases in which n is coprime to 12, i.e., n ≡ 1, 5, 7, or 11 (mod 12). In these cases, 12 has a unique multiplicative inverse modulo n, call it k, whence 12x ≡ 8 (mod n) has the unique solution x ≡ 8k (mod n).

Next, consider the cases in which gcd (n, 12) = 2, i.e., n ≡ 2 or 10 (mod 12). Let m = n/2. ...

#### Solution Summary

In this solution we investigate the solutions of the congruence 12x ≡ 8 (mod n) for various values of n.

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