Problem #55 EASY
The Antique Clock Shop
Scenario Number Theory Logic Pattern Recognition
Problem Statement
In a narrow lane in the old city of Hyderabad, there is a small shop owned by a clockmaker named Rustam. He has been repairing and collecting antique clocks for fifty years. Every clock in his shop runs perfectly.
One afternoon, a customer walks in and hears something delightful: all of the clocks in the shop are chiming at the same moment. Rustam explains that this happens once every so often, when all the clocks' chiming intervals align.
The customer looks around. There are three grandfather clocks. The first chimes every 15 minutes. The second chimes every 20 minutes. The third chimes every 30 minutes.
'They all just chimed together right now,' Rustam says, glancing up from a gear he is polishing. 'When will they all chime together again?'
The customer thinks for a moment, then smiles.
The three clocks chime at intervals of 15, 20, and 30 minutes respectively. They all chime together right now. How many minutes from now will they next all chime simultaneously?
Answer & Quick Explanation
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The three clocks will next all chime together in exactly 60 minutes. LCM(15, 20, 30) = 60. At that point, Clock A has chimed 4 times, Clock B has chimed 3 times, and Clock C has chimed 2 times since the last simultaneous chime.
Detailed Editorial Solution
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When multiple periodic events need to coincide again, the answer is always the Lowest Common Multiple (LCM) of their periods. The LCM is the smallest number that is divisible by all given periods simultaneously.
Step 1: Clock A chimes every 15 minutes. Clock B chimes every 20 minutes. Clock C chimes every 30 minutes.
Step 2: Prime factorise each interval: 15 = 3 x 5. 20 = 2² x 5. 30 = 2 x 3 x 5.
Step 3: LCM = product of the highest power of each prime that appears in any factorisation.
Step 4: Primes involved: 2, 3, 5. Highest powers: 2² (from 20), 3¹ (from 15 or 30), 5¹ (from all three).
Step 5: LCM = 2² x 3 x 5 = 4 x 3 x 5 = 60.
Step 6: Verify: 60/15 = 4 (Clock A chimes 4 times), 60/20 = 3 (Clock B chimes 3 times), 60/30 = 2 (Clock C chimes 2 times). All are whole numbers — all three chime at the 60-minute mark.
Key Insight:
LCM by prime factorisation is a universally applicable technique. The key step that people often miss is taking the HIGHEST power of each prime across all numbers, not just the primes that appear in all numbers (that would give GCD, not LCM). LCM problems appear naturally whenever periodic events need to synchronise.