1. **State the problem:** We need to find the greatest possible cost for a single ticket given two total prices for groups of tickets: ₱975 and ₱1170. 2. **Understand the problem:** Each price represents a total amount paid for some number of tickets, all the same cost in the respective group. 3. **Identify the mathematical approach:** The ticket cost must be a common divisor of both total amounts (₱975 and ₱1170). The ticket cost is the greatest common divisor (GCD) of 975 and 1170. 4. **Find the prime factorizations:** - 975 = 3 x 5² x 13 - 1170 = 2 x 3 x 5 x 3 x 13 (or 2 x 3² x 5 x 13) 5. **Find the GCD:** Take the minimum powers of each common prime: - For 2: appears only in 1170 (skip) - For 3: min power is 1 - For 5: min power is 1 - For 13: min power is 1 Therefore, GCD = 3 x 5 x 13 = 195. 6. **Interpretation:** The highest possible cost of a single ticket that fits both group totals is ₱195. **Final answer:** The most a ticket could cost is **₱195**.