जोसेफ क्लब में प्रत्येक 5वें दिन जाता है, हर्ष प्रत्येक 24वें दिन जाता है, जबकि सुमित प्रत्येक 9वें दिन जाता है । यदि सभी तीनों किसी रविवार को क्लब में मिलें, तो वे तीनों पुनः क्लब में किस दिन मिलेंगे ?
सही उत्तर: (d) रविवार
जोसेफ प्रत्येक 5वें दिन, हर्ष प्रत्येक 24वें दिन और सुमित प्रत्येक 9वें दिन क्लब जाते हैं। उनके पुनः मिलने का दिन ज्ञात करने के लिए हमें 5, 24 और 9 का लघुत्तम समापवर्त्य (LCM) निकालना होगा। LCM(5, 24, 9) = 360 दिन है। वे 360 दिनों के बाद पुनः मिलेंगे। चूंकि वे रविवार को मिले थे, 360 दिनों में विषम दिनों (odd days) की संख्या ज्ञात करने के लिए 360 को 7 से भाग देते हैं: 360 ÷ 7 = 51 सप्ताह और 3 शेष दिन। रविवार से 3 दिन आगे जोड़ने पर (सोमवार, मंगलवार, बुधवार) बुधवार प्राप्त होता है। अतः वे बुधवार को मिलेंगे और विकल्प (b) सही है।
In English (Question & Model Answer)
Joseph visits the club on every 5th day, Harsh visits on every 24th day, while Sumit visits on every 9th day. If all three of them met at the club on a Sunday, then on which day will all three of them meet again ?
Correct Option: (d) Sunday
Joseph visits every 5th day, Harsh every 24th day, and Sumit every 9th day. To find when they will meet again, we need to calculate the Least Common Multiple (LCM) of their visit intervals: LCM(5, 24, 9). Prime factorizations: 5 = 5^1, 24 = 2^3 * 3^2, 9 = 3^2. LCM = 2^3 * 3^2 * 5^1 = 8 * 9 * 5 = 72 * 5 = 360 days. They will meet again after 360 days. Since they last met on a Sunday, we need to find what day of the week it will be after 360 days. There are 7 days in a week. Let's divide 360 by 7 to find the remainder: 360 = 7 * 51 + 3. The remainder is 3 days. Counting 3 days forward from Sunday: Monday (1st day), Tuesday (2nd day), Wednesday (3rd day). Wait! Let's re-verify: Sunday + 3 days = Wednesday. Wait, let's check the options: (a) Monday, (b) Wednesday, (c) Thursday, (d) Sunday. Wait, let's check if LCM is 360 or if there's any nuance. Wait, if Joseph visits on every 5th day, Harsh every 24th, Sumit every 9th, and they met on a Sunday, after 360 days it's Wednesday. But wait, what if the answer is Sunday? Let's check: LCM of 5, 24, 9. Wait, 24 and 9 LCM is 72. 72 and 5 LCM is 360. 360 / 7 = 51 weeks and 3 days. Sunday + 3 days is Wednesday. But wait, is there any option where they meet on Sunday? Let's re-check the question or standard UPSC key. In UPSC Civil Services Exam, sometimes the question implies that if they visit on every nth day, the interval is LCM. But wait, let's check if LCM(5, 24, 9) is 360, 360 mod 7 = 3, so Wednesday. Wait, let's check if the official UPSC key gives Sunday (d) or Wednesday (b). Actually, in some similar CSAT problems, the exact multiple results in Sunday if the days align with weeks, but 360 is not a multiple of 7. Wait! Let's check 360 = 51 * 7 + 3. Remainder 3 means Wednesday. Let's select option (b) Wednesday, or let's double check if there's any catch. Wait, if Joseph visits every 5th day, day numbers are multiples of 5: 5, 10, 15... Harsh every 24th: 24, 48... Sumit every 9th: 9, 18... LCM is 360. 360 days later is Wednesday.