1. The problem asks about the usefulness of the LCM (Least Common Multiple) of 12 and 15. 2. The LCM of two numbers is the smallest number that is a multiple of both. 3. This is useful for finding when two events will happen together again, as it represents the first time both cycles align. 4. Therefore, the correct answer is C: Finding when two events will happen together again. 5. Next, we find the largest number of students per group possible when there are 36 boys and 48 girls. 6. This is the Greatest Common Divisor (GCD) problem, as groups must have equal numbers of boys and girls. 7. Find the GCD of 36 and 48: - Prime factors of 36: $2^2 \times 3^2$ - Prime factors of 48: $2^4 \times 3$ - GCD is the product of the lowest powers: $2^2 \times 3 = 4 \times 3 = 12$ 8. So, the largest number of students per group is 12. 9. The correct answer is A: 12. 10. Finally, find after how many seconds two traffic lights changing every 30 and 45 seconds will change simultaneously. 11. This is an LCM problem: find LCM of 30 and 45. 12. Prime factors: - 30 = $2 \times 3 \times 5$ - 45 = $3^2 \times 5$ 13. LCM takes the highest powers: $2 \times 3^2 \times 5 = 2 \times 9 \times 5 = 90$ 14. So, both lights will change together after 90 seconds. 15. The correct answer is B: 90.