LCM and HCF.pptx problems and didferent models

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📑 Outline of the 3000-Word Write-up on Remainders

1. Introduction to Remainders


2. Basic Concepts and Definitions

Dividend, Divisor, Quotient, and Remainder

Properties of Division



3. Rules and Properties of Remainders

Addition, Subtraction, Multiplication with Remainders

Division and Powers with Remainders



4. Special Theorems

Remainder Theorem (Algebra)

Fermat’s Little Theorem

Euler’s Theorem

Wilson’s Theorem

Chinese Remainder Theorem



5. Shortcut Techniques for Competitive Exams

Finding last digits

Remainders in factorials

Remainders with large powers



6. Step-by-Step Solved Examples

Easy, Moderate, and Advanced Levels



7. Applications of Remainders

Aptitude Exams (CAT, GATE, TCS NQT, etc.)

Computer Science (modulo operations, hashing)

Cryptography



8. Common Mistakes and Misconceptions


9. Practice Questions with Solutions


10. Conclusion and Tips




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1. 8/15/2025 1 Department of CSSP, SR University Aptitude Training Team Department of CSSP SR University Course Code Course Title L R P Cr 23HS102CR202 QALR 1 4 3 Career Readiness Elective Unit 2 – Number System Topic LCM-HCF

2. 8/15/2025 2 Slide Title L.C.M of Numbers by division Method

3. 8/15/2025 3 Slide Title L.C.M of Numbers Prime factorization method

4. 8/15/2025 4 Slide Title H.C.F of numbers

5. 8/15/2025 5 Slide Title Relationship Between LCM and HCF

6. 8/15/2025 6 Slide Title LCM 1. Find the LCM of 15, 24, 35 and 54. a. 7560 b. 5670 c. 7650 d. 6570 2. Find the LCM of 48 , 72 , 140 a.5000 b.5060 c. 5040 d. none

7. 8/15/2025 7 Slide Title LCM 3. Find the LCM of 2, 3, 4, 5, 6, 7, 8, 9 a.2500 b.2520 c. 2200 d. none 4. Find the LCM of 32,36,40 a.1400 b. 1440 c.1435 d. none

8. 8/15/2025 8 Slide Title HCF 5. Find the HCF of 30 ,42 ,135 a. 3 b. 5 c. 6 d. 8 6. Find the HCF of 210 , 495 , 980 a. 3 b. 7 c. 5 d. none

9. 8/15/2025 9 Slide Title HCF 7. Find the HCF of 639 , 1065 , 1491 a. 426 b. 313 c. 213 d. 216

10. 8/15/2025 10 Slide Title LCM AND HCF 8. Find the LCM of 7, , a. b. c. d.  

11. 8/15/2025 11 Slide Title LCM AND HCF 9. Find the HCF of 7, , a. b. c. d.  

12. 8/15/2025 12 Slide Title LCM AND HCF 9. Find the HCF of 7, , a. b. c. d.  

13. 8/15/2025 13 Slide Title LCM AND HCF (Fractions) 10. Find the LCM of 3/2, 81/16, 9/8 and 9/8 a. 81/2 b. 91/2 c. 101/2 d. 111/2

14. 8/15/2025 14 Slide Title LCM AND HCF (Fractions) 11. Find the HCF of 4/3 , 4/9 , 2/15 , 36/21 a. 2/315 b. 3/215 c. 2/154 d. 2/317

15. 8/15/2025 15 Slide Title LCM AND HCF (decimals) 18. What is the LCM of 0.15, 0.18 and 0.45? a. 0.81 b. 0.6 c. 0.09 d. 0.9

16. 8/15/2025 16 Slide Title LCM AND HCF (Relation) 19. The product of two numbers is 726 and their HCF is 11, then their LCM is: a. 58 b. 68 c. 66 d. 76

17. 8/15/2025 17 Slide Title LCM AND HCF (Relation) 20. Two numbers are in the ratio 3: 4. The product of their HCF and LCM is 2700. The sum of the numbers is: a. 105 b. 45 c. 60 d. 15

18. 8/15/2025 18 Slide Title LCM AND HCF (Relation) 21.The LCM of two numbers is 660 and their HCF is 5. If one of the numbers is 55, find the other. a. 12 b. 110 c. 60 d. 275

19. 8/15/2025 19 Slide Title LCM AND HCF (Relation) 22. Three numbers are in the ratio of 5:7:9 and their LCM is 34,650. Their HCF is: a. 315 b. 55 c. 99 d. 110

20. 8/15/2025 20 Slide Title LCM AND HCF (Relation) 23. The LCM of two numbers is 15 times of their HCF . If product of those numbers is 14415 , then find their HCF a. 38 b. 25 c. 18 d. 31

21. 8/15/2025 21 Slide Title LCM AND HCF (Relation) 24. The LCM of two numbers is 45 times of its HCF and sum of HCF and LCM is 1150. If one of the number is 125 then find the second number ? a. 325 b.225 c. 175 d. 215

22. 8/15/2025 22 Slide Title LCM AND HCF (Relation) 25. The sum of two numbers is 216 and their HCF is 27. How many pairs of such numbers can be formed ? a. 1 b. 2 c. 3 d. 4

23. 8/15/2025 23 Slide Title LCM AND HCF (Relation) 26. The sum of two numbers is 1215 and their HCF is 81 . How many pairs of such numbers can be formed? a. 1 b. 2 c. 3 d. 4

24. 8/15/2025 24 Slide Title LCM AND HCF 27. What is the least possible number which when divided by 8 , 12 and 24 in each case it leaves the remainder 7 ? a. 28 b. 31 c. 27 d. 35

25. 8/15/2025 25 Slide Title LCM AND HCF 28. What is the least possible number when divided by 18 , 35 , 42 it leaves 2 , 19 , 26 as the remainders respectively ? a. 620 b. 636 c. 614 d. 610

26. 8/15/2025 26 Slide Title LCM AND HCF 29. In a large school auditorium, the students are mad to sit to watch the program. If the teachers make a row of students of 16 each , there will be 12 students left . If they make row of 24 each ,then there will be 20 students left, if they make rows of 25 each, there will be 21 students left and if they make rows of 30 each , there will be 26 students left. What is the minimum number of students present in the school ? a.1216 b.1784 c.1196 d.2396

27. 8/15/2025 27 Slide Title LCM AND HCF 30. What is the least possible number which when divided by 13 leaves the remainder 3 and when divided by 5 it leaves the remainder 2. a. 38 b. 40 c. 52 d. 42

28. 8/15/2025 28 Slide Title LCM AND HCF 31. what is the largest four digit number which is divisible by 4 ,5 , 8 and 10. a.9960 b. 9940 c. 9980 d. 9990

29. 8/15/2025 29 Slide Title LCM AND HCF 32. find the smallest number of five digits exactly divisible by 6 ,8 ,10 and 15 a. 10040 b. 10080 c. 10120 d. 10160

30. 8/15/2025 30 Slide Title LCM AND HCF 33. four bells ring at 10,12,15 and 20 min respectively beginning together. After what interval of time do they ring again together ? a. 50 min b. 80 min c. 100 min d. 60 min

31. 8/15/2025 31 Slide Title LCM AND HCF 34. Find the largest possible number with which when 62 and 100 are divided it leaves the remainder 5 in each case ? a. 15 b. 19 c.21 d.13

32. 8/15/2025 32 Slide Title LCM AND HCF 35. Find the greatest possible number with which when we divide 37 and 58 , it leaves the respective remainders of 2 and 3. a. 15 b. 20 c. 5 d. 18

33. 8/15/2025 33 Slide Title LCM AND HCF 36. Find the largest possible number with which when 40 , 68 and 82 are divided the remainders remains the same. a. 24 b. 13 c. 14 d. 10

34. 8/15/2025 34 Slide Title LCM AND HCF 37. Find the largest possible number with which when 39 , 65 and 91 are divided leaves same remainders in each case. a. 26 b. 36 c. 28 d. 30

35. 8/15/2025 35 Slide Title LCM AND HCF 38. What is the greatest possible length of a scale , that can be used to measure exactly three blocks of wood of lengths 2 m 40 cm , 3 m 36 cm and 4 m 32 cm ? a. 12 cm b. 96 cm c. 48 cm d. 24 cm

36. 8/15/2025 36 Slide Title LCM AND HCF 39. In a seminar , the number of participants in Hindi ,English and mathematics are 60 ,84 and 108 respectively. Find the minimum number of rooms required if in each room the same number of participants are to be seated and all of them being in the same subject ? a. 34 b. 25 c. 21 d. 15

37. 8/15/2025 37 Slide Title LCM AND HCF 40. Three numbers are in the ratio 35:55:77 , the HCF of those numbers is 21. what is the smallest number of the three ? a. 245 b. 105 c. 735 d. 280

38. 8/15/2025 38 Slide Title LCM AND HCF 41. A and B are two prime numbers such that A>B and their LCM is 209. the value of - B is: a. 350 b. 372 c. 361 d. 339  

39. 8/15/2025 39 Slide Title LCM AND HCF 42. The LCM of two prime numbers x and y (x > y) is 533. The value of 4y-x is: a. 11 b. 18 c. 23 d. 21

40. 8/15/2025 40 Slide Title LCM AND HCF