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+### Statement
+
+$5.5.1.$ [Insert the problem statement]
+
+### Solution
+
+### Problem
+
+$5.5.1.$ The volume of a gas was halved, and its temperature was increased by a factor of $1.5$. By what factor did the gas pressure increase?
+
+### Solution
+
+The amount of gas in this process remains constant, so for the two states (index 1 for the initial state, index 2 for the final state), we can write the ideal gas equation of state:
+
+$$
+\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}
+$$
+
+According to the problem statement, we are given: $V_2 = \frac{1}{2}V_1$ and $T_2 = \frac{3}{2}T_1$.
+
+Substituting these into the equation:
+
+$$
+\frac{P_1V_1}{T_1} = \frac{P_2 \cdot \frac{1}{2}V_1}{\frac{3}{2}T_1}
+$$
+
+Hence:
+
+$$
+P_2 = 3P_1
+$$
+
+#### Answer:
+
+By a factor of $3$
+
+#### Answer
+
+[Insert a concise answer or boxed result]