### 13 Mar

Arithmetic is considered the mainstay of Quantitative Aptitude section in competitive exams such SSC CGL and Bank PO. Questions based on LCM are among the most scoring ones within arithmetic.

Vidya Guru, the **centre for Best SSC Coaching in Delhi / NCR**, has provided below a few quality word problems on LCM. They will enhance your conceptual clarity and help you get a better grip on the topic. Thus, we suggest that you attempt each of them.

**Basic Level**

**Question-1.** Mary and John are both practicing for a math competition. Mary practices every 5 days, and John practices every 7 days. If they both started practicing on the same day, on what day will they both be practicing again on the same day?

**Solution:** To find the day when they will both practice on the same day, we need to find the LCM of 5 and 7. The multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, … and the multiples of 7 are: 7, 14, 21, 28, 35, 42, 49, … The smallest number that appears in both lists is 35, so they will both be practicing on the same day 35 days after they started.

Question-2. A factory produces three types of products. Product A is produced every 4 days, product B is produced every 6 days, and product C is produced every 8 days. If the factory produces all three products on the same day, how long will it take for them to produce all three products on the same day again?

**Solution:** To find the day when all three products will be produced on the same day again, we need to find the LCM of 4, 6, and 8. The multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, … the multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, … and the multiples of 8 are: 8, 16, 24, 32, 40, 48, … The smallest number that appears in all three lists is 24, so it will take 24 days for the factory to produce all three products on the same day again.

**Question-3.** Sarah and Tom are painting a wall together. Sarah can paint the wall in 12 hours, and Tom can paint the wall in 16 hours. If they work together to paint the wall, how long will it take them to finish?

**Solution:** To find the time it takes for them to finish painting the wall together, we need to find the LCM of 12 and 16. The multiples of 12 are: 12, 24, 36, 48, 60, 72, 84, 96, … and the multiples of 16 are: 16, 32, 48, 64, 80, 96, … The smallest number that appears in both lists is 48, so it will take them 48/3 = 16 hours to finish painting the wall together, where we divide by 3 as both workers work together.

**Higher Level**

**Question-4.** Three cyclists start riding from a point at the same time. The first cyclist rides every 6 minutes, the second cyclist rides every 8 minutes, and the third cyclist rides every 10 minutes. After how many minutes will they meet at the starting point again?

**Solution:** We need to find the LCM of 6, 8, and 10. The multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, … the multiples of 8 are: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, … and the multiples of 10 are: 10, 20, 30, 40, 50, 60, 70, 80, … The smallest number that appears in all three lists is 120. Therefore, they will meet at the starting point again after 120 minutes.

**Question-5.** A train starts from station A and another train starts from station B at the same time. The first train travels at a speed of 40 km/h and the second train travels at a speed of 60 km/h. The distance between the two stations is 600 km. After how much time will the two trains meet?

**Solution:** The two trains are moving towards each other, so their relative speed is the sum of their speeds, which is 40 km/h + 60 km/h = 100 km/h. The time taken to cover the distance between the two stations is equal to the distance divided by the relative speed, which is 600 km/100 km/h = 6 hours.

**Question-6.** A factory produces three types of products. Product A is produced every 5 days, product B is produced every 7 days, and product C is produced every 9 days. If the factory produces all three products on the same day, how long will it take for them to produce all three products on the same day again?

**Solution:** We need to find the LCM of 5, 7, and 9. The multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, … the multiples of 7 are: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, … and the multiples of 9 are: 9, 18, 27, 36, 45, 54, 63, 72, … The smallest number that appears in all three lists is 315. Therefore, it will take 315 days for the factory to produce all three products on the same day again.

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