The Antique Clock Shop

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.

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.