Six bells start together and ring at intervals of 2 s, 4 s, 6 s, 8 s, 10 s, and 12 s. In 30 minutes, how many times will they ring together (including the start)?

Introduction / Context:
When multiple periodic events begin together, they will next coincide at multiples of the least common multiple (LCM) of their intervals. Counting coincidences over a duration includes the initial coincidence at t = 0 if asked to include the start.

Given Data / Assumptions:

• Intervals: 2 s, 4 s, 6 s, 8 s, 10 s, 12 s
• Duration: 30 minutes = 1800 s
• Include the initial start together

Concept / Approach:
Find LCM of the intervals to get the repeat time for simultaneous ringing. Then the count of coincidences including the start is floor(1800 / LCM) + 1.

Step-by-Step Solution:
Prime factors: 2, 4 = 2^2, 6 = 2*3, 8 = 2^3, 10 = 2*5, 12 = 2^2*3.LCM = 2^3 * 3 * 5 = 8 * 3 * 5 = 120 s.Number of coincidences = floor(1800 / 120) + 1 = 15 + 1 = 16.

Verification / Alternative check:
They coincide at t = 0, 120, 240, …, 1800. That’s 16 equally spaced times including t = 0 and t = 1800.

Why Other Options Are Wrong:
4, 10, 14, 15 arise from excluding the start or miscomputing the LCM or number of intervals within 1800 s.

Common Pitfalls:
Forgetting to include the start as instructed; using gcd instead of LCM; arithmetic slips in 1800/120.

Final Answer:
16