Shortcut Trick The time after which all bells ring together is the LCM of their individual ringing intervals. LCM of 6, 7, 9, and 12 = 252 seconds. Since the bells ring together every 252 seconds, we check for 252 or its multiples in the options. Next multiple = 252 × 2 = 504 seconds, which matches an option. ∴ The correct answer is 504 seconds. Alternate Method Given: Intervals at which the four bells ring = 6, 7, 9, and 12 seconds. Concept Used: The bells will ring together again after a time that is exactly divisible by all the individual intervals. This specific time represents the Least Common Multiple (LCM) of the given numbers. If the exact LCM is not among the options, any integer multiple of the LCM is a valid answer since the bells continue to ring together periodically. Calculations: ⇒ Prime factorization of 6 = 2 × 3 ⇒ Prime factorization of 7 = 7 ⇒ Prime factorization of 9 = 3^(2) ⇒ Prime factorization of 12 = 2^(2) × 3 ⇒ LCM = Product of the highest powers of all present prime factors ⇒ LCM = 2^(2) × 3^(2) × 7 ⇒ LCM = 4 × 9 × 7 = 252 seconds ⇒ This means the bells will ring together every 252 seconds (i.e., at 252s, 504s, 756s, etc.). ⇒ Analyzing the given options, we find that 504 is exactly divisible by 252 (252 × 2 = 504). ∴ The correct answer is 504 seconds. Related Concepts: LCM for Periodic Events For events that repeat at different regular intervals (like bells tolling, traffic lights changing, or runners on a circular track), they will all occur together again at exactly the LCM of their individual intervals. Multiples of LCM If the events coincide for the first time at the LCM, they will continue to simultaneously coincide at every successive integer multiple of the LCM (e.g., 2 × LCM, 3 × LCM). HCF for Maximum Grouping While LCM is for simultaneous forward occurrences, the Highest Common Factor (HCF) is used when dividing items into the largest possible equal groups without leaving any remainder (e.g., cutting ropes into maximum equal lengths).