Given that $x$ is a positive integer less than 100, how many solutions does the congruence $x + 13 \equiv 55 \pmod{34}$ have?
I want to find all the values of $x$ that satisfy the congruence. I can rewrite the congruence as $x \equiv 55 - 13 \pmod{34}$. This simplifies to $x \equiv 42 \pmod{34}$. This means that $x$ has the same remainder as 42 when divided by 34. One way to find such values of $x$ is to add or subtract multiples of 34 to 42. For example, $x = 42 - 34 = 8$ is one solution. Another solution is $x = 42 + 34 = 76$. I can keep adding or subtracting 34 to 42 until I reach the boundaries of the given range. The smallest possible value of $x$ is $42 - 2 \cdot 34 = -26$. The largest possible value of $x$ is $42 + 2 \cdot 34 = 110$. However, I only want positive integers less than 100, so I can ignore the values outside that range. The solutions in the range are $x = 8, 42, 76$. There are three solutions in total. 