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1. All we need to do here is divide both sides by 7. We add 13 to the right-hand side repeatedly until we get a multiple of 7 (adding 13 is the same as adding 0, so this is legal). We get \(25\text{,}\) \(38\text{,}\) \(51\text{,}\) \(64\text{,}\) \(77\) – got it. So we have:

   \begin{equation*} \begin{aligned}7x \amp \equiv 12 \pmod{13} \\ 7x \amp \equiv 77 \pmod{13} \\ x \amp \equiv 11 \pmod{13}. \end{aligned} \end{equation*}

2. Here, since we have numbers larger than the modulus, we can reduce them prior to applying any algebra. We have \(84 \equiv 9\text{,}\) \(38 \equiv 8\) and \(79 \equiv 4\text{.}\) Thus,

   \begin{equation*} \begin{aligned}84x - 38 \amp \equiv 79 \pmod{15} \\ 9x - 8 \amp \equiv 4 \pmod{15} \\ 9x \amp \equiv 12 \pmod{15} \\ 9x \amp \equiv 72 \pmod{15}. \end{aligned} \end{equation*}

   We got the 72 by adding \(0 \equiv 60 \pmod{15}\) to both sides of the congruence. Now divide both sides by 9. However, since \(\gcd(9, 15) = 3\text{,}\) we must divide the modulus by 3 as well:

   \begin{equation*} x \equiv 8 \pmod 5\text{.} \end{equation*}

   So the solutions are those values which are congruent to 8, or equivalently 3, modulo 5. This means that in some sense there are 3 solutions modulo 15: 3, 8, and 13. We can write the solution:

   \begin{equation*} x \equiv 3 \pmod{15}; ~~ x \equiv 8 \pmod{15}; ~~x \equiv 13 \pmod{15}\text{.} \end{equation*}

3. First, reduce modulo 14:

   \begin{equation*} 20x \equiv 23 \pmod{14} \end{equation*}
   \begin{equation*} 6x \equiv 9 \pmod{14}\text{.} \end{equation*}

   We could now divide both sides by 3, or try to increase 9 by a multiple of 14 to get a multiple of 6. If we divide by 3, we get,

   \begin{equation*} 2x \equiv 3 \pmod{14}\text{.} \end{equation*}

   Now try adding multiples of 14 to 3, in hopes of getting a number we can divide by 2. This will not work! Every time we add 14 to the right side, the result will still be odd. We will never get an even number, so we will never be able to divide by 2. Thus there are no solutions to the congruence.