LCM Examples for Beginners: 12 Solved Problems

Reading about LCM is useful, but nothing builds confidence like solving problems. This collection gives you twelve solved LCM examples arranged from very easy to exam level, using Indian settings such as school timetables, local trains, markets and festivals. Try each question yourself before reading the solution, and use the free LCM calculator afterwards to check. Each example also tells you which idea it is practising so that you know what to revise.

Level 1: Warm-Up Examples with Small Numbers

Example 1: LCM of 3 and 4. Multiples of 3: 3, 6, 9, 12. Multiples of 4: 4, 8, 12. The first match is 12. Answer: 12. Idea: listing multiples.

Example 2: LCM of 5 and 10. Since 10 is a multiple of 5, the LCM is 10. Answer: 10. Idea: one number divides the other.

Example 3: LCM of 7 and 9. They share no factor except 1, so the LCM is the product, 63. Answer: 63. Idea: co-prime numbers.

Example 4: LCM of 6, 8 and 12. Multiples of 12: 12, 24. Check 24 against 6 (4 times) and 8 (3 times). Answer: 24. Idea: start from the biggest number.

Level 2: Prime Factor and Ladder Practice

Example 5: LCM of 24 and 36. 24 = 2^3 x 3 and 36 = 2^2 x 3^2. Highest powers are 2^3 and 3^2, so LCM = 8 x 9 = 72. Answer: 72.

Example 6: LCM of 15, 20 and 35. Ladder method: divide by 5 to get 3, 4, 7. They have no common prime pair, so LCM = 5 x 3 x 4 x 7 = 420. Answer: 420.

Example 7: LCM of 18, 27 and 45. 18 = 2 x 3^2, 27 = 3^3, 45 = 3^2 x 5. Highest powers: 2, 3^3, 5. LCM = 2 x 27 x 5 = 270. Answer: 270.

Level 3: Indian Word Problems

Example 8: Local trains. On a Mumbai suburban line, one train leaves a station every 12 minutes and another every 15 minutes. Both leave at 7:00 AM. When do they next leave together? LCM(12, 15) = 60 minutes, so at 8:00 AM. Answer: 8:00 AM.

Example 9: Sweets for Diwali. A shop packs laddoos in boxes of 8 and barfi in boxes of 12. What is the least number of each that gives equal counts? LCM(8, 12) = 24, so 3 boxes of laddoos and 2 boxes of barfi, each giving 24 pieces. Answer: 24 pieces each.

Example 10: Smallest number with remainders. Find the smallest number which leaves remainder 3 when divided by 6, 8 and 12. LCM(6, 8, 12) = 24, so the number is 24 + 3 = 27. Answer: 27. This pattern, LCM plus the common remainder, appears often in bank and SSC papers.

Example 11: Largest number to add. What is the smallest number that must be added to 100 so that it becomes divisible by 12, 15 and 20? LCM = 60. The next multiple of 60 above 100 is 120, so add 20. Answer: 20.

Example 12: Fractions. Add 5/12 and 7/18. LCM(12, 18) = 36, so 15/36 + 14/36 = 29/36. Try it on a fraction calculator to confirm. Answer: 29/36.

Key takeaway: Most word problems reduce to one of four patterns: meeting again, equal packs, number plus a common remainder, or common denominators for fractions.

How to Build Your Own Practice Set

Once you have finished these twelve problems, create a personal worksheet. Pick pairs of numbers from your own life: the number of students in two tuition batches, the days between two recurring bills, or the minutes between two bus services on your route. Find the LCM by one method, then switch to a second method to confirm. If both methods agree, you have almost certainly got it right. If they differ, find the slip, because that is where the learning happens. Parents can help by turning the task into a small game at home, asking the child to predict when two repeating things, like a sibling visit and a weekly market day, will fall on the same date. This kind of practice makes LCM feel useful instead of abstract, and it prepares students for the scheduling style questions that appear in many competitive exams.

Bonus Challenge Examples with Hints

Challenge 1: Four bells ring at intervals of 10, 15, 20 and 25 seconds. If they ring together at the start of a school assembly, after how many seconds will they ring together again? Hint: factor each number and take the highest powers, 2^2, 3 and 5^2. LCM = 4 x 3 x 25 = 300 seconds, which is 5 minutes.

Challenge 2: What is the least number of Rs. 5, Rs. 10 and Rs. 20 coins that, in equal value, make the smallest common amount? Hint: you want the smallest amount that is a multiple of 5, 10 and 20. LCM = 20, so Rs. 20 can be made from 4 five-rupee coins, 2 ten-rupee coins or 1 twenty-rupee coin.

Challenge 3: Two numbers are in the ratio 3 : 4 and their LCM is 240. Find the numbers. Hint: write them as 3k and 4k. Their LCM is 12k, so 12k = 240 and k = 20. The numbers are 60 and 80. Ratio plus LCM questions like this one appear often in bank exams, and the trick of writing numbers as multiples of a common factor k solves almost all of them.

Benefits of Practising with Solved Examples

Working through solved examples trains pattern recognition. After a few problems you will spot instantly whether a question wants LCM or HCF. You also learn the typical phrases used by exam setters, such as “at the same time again”, “smallest number divisible by” and “least number of items”, which all point to LCM. Regular practice improves speed, and speed matters in exams like SSC CGL and bank clerk tests, where you have barely a minute per question.

Challenges Beginners Run Into

Beginners often struggle with the jump from simple lists to prime factors because it feels like a new topic. Word problems add a language layer on top of the arithmetic. Remainder questions are particularly confusing, as they combine LCM with a small addition. The fix is to practise one pattern at a time and keep a short notebook of each pattern with one model solution.

Common Mistakes in Examples Like These

1. Adding the remainder before taking LCM. Always find LCM first, then add the common remainder.
2. Using the product for non co-prime numbers. 8 x 12 = 96 is a common multiple but not the least; the LCM is 24.
3. Using HCF for scheduling questions. If events must meet again, you need LCM.
4. Arithmetic slips in the ladder. Re-divide each row carefully, since one wrong quotient carries down.
5. Forgetting leftover numbers in the ladder product. Multiply divisors and remaining numbers.
6. Ignoring the question’s final ask. In Example 11 the answer is the number to add, 20, not the LCM or 120.

Best Practices for Beginners

1. Do five questions a day for two weeks rather than fifty in one day.
2. Write the method name next to each solution so you build a toolkit.
3. Always verify by dividing your answer by each input.
4. Time yourself once a week, aiming under one minute for two-number problems.
5. Check difficult answers on the LCM calculator, then redo the working to see where you differed.
6. Create your own word problem from your daily life, like bus or tuition timings.

Conclusion

These twelve examples cover the full range from listing multiples to remainder problems. Keep them as a revision sheet, and extend it with your own questions. When you can explain why each answer works, you have moved beyond memorising and are ready for any LCM question that school or competitive exams may throw at you.

FAQs

How do I practise LCM as a beginner?
Start with small numbers using listing, then move to prime factors and the ladder method. Solve five to ten problems a day and check each with a calculator.

What is the LCM of 12 and 15?
The multiples of 12 are 12, 24, 36, 48, 60 and of 15 are 15, 30, 45, 60. The first match is 60, so the LCM is 60.

How do I find the smallest number that leaves the same remainder?
Find the LCM of the divisors and add the common remainder. For divisors 6, 8 and 12 with remainder 3, the answer is 24 + 3 = 27.

How do I know whether a problem needs LCM or HCF?
If the question is about events meeting again, smallest common quantity or common denominators, use LCM. If it is about cutting or sharing into the largest equal parts, use HCF.

Is there a quick way to check my LCM answer?
Divide your answer by every given number and confirm there is no remainder. For two numbers you can also verify that LCM x HCF equals their product.