The traffic lights at three different road crossings change after every 48 sec., 72 sec. and 108 sec. respectively. If they all change simultaneously at 8:20:00 hours, then at what time will they again change simultaneously?

### Concept & Strategy When periodic events (like bells ringing, lights changing, or people running a track) occur at different intervals, they will all happen together again at a time interval equal to the Lowest Common Multiple (L.C.M.) of their individual periods. $$ \text{Simultaneous Interval} = \text{L.C.M.}(T_1, T_2, T_3, \dots) $$ ### Step-by-Step Solution * **Given:** Intervals of change are 48 sec, 72 sec, and 108 sec. Initial simultaneous change was at 8:20:00. * Find the L.C.M. of 48, 72, and 108 using prime factorization: * 48 = $2^4 \times 3$ * 72 = $2^3 \times 3^2$ * 108 = $2^2 \times 3^3$ * Extract the highest powers of the prime factors: * Highest power of 2 is $2^4 = 16$. * Highest power of 3 is $3^3 = 27$. * Multiply them to get the L.C.M.: * L.C.M. = 16 $\times$ 27 = 432 seconds. * This means the lights will change together again after 432 seconds. * Convert 432 seconds into minutes and seconds: * 432 / 60 = 7 minutes with a remainder of 12 seconds. * So, 432 seconds = 7 minutes 12 seconds. * Add this time interval to the initial time (8:20:00): * 8 hours : 20 minutes : 00 seconds * + 0 hours : 7 minutes : 12 seconds * = 8:27:12 hours. ### Exam Strategy & Shortcut Instead of heavy prime factorization, look for common large factors to speed up L.C.M. calculation. All numbers are multiples of 12. * 48 = 12 $\times$ 4 * 72 = 12 $\times$ 6 * 108 = 12 $\times$ 9 The L.C.M. of (4, 6, 9) is simply 36. Multiply the common factor back: 12 $\times$ 36 = 432 seconds. Then convert to minutes rapidly. ### Common Pitfall A careless mistake in time addition. Students calculate 432 seconds correctly but mess up the base-60 conversion (e.g., assuming 100 seconds in a minute), leading to an incorrect final timestamp. Always meticulously divide by 60 for time. ### Final Answer **Therefore, the correct answer is 8:27:12 hrs.**