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Problems on H.C.F and L.C.M: HCF (Highest Common Factor) and LCM (Least Common Factor) are the basics of many mathematical operations. These help solve the various complex mathematical problems at a wide variety of competitive exams like RBI Assistant Exam, GATE, NEET, and more. To ace the performance at the examination, it is important to develop a grip with these questions. Scroll below to find practice problems on H.C.F. and L.C.M.
Table of Contents
Formulas to Find HCF and LCM
For the problems on H.C.F and L.C.M, there are different ways to find them individually. Let’s explore them one by one:
Division Method
You need to follow three steps to find the HCF by the division method. The steps are below.
- Step 1- Divide the larger numbers by smaller numbers.
- Step 2- You will get a remainder from Step 1. Assume this remainder as the divisor, and then the last divisor will be taken as the dividend.
- Step 3- You need to repeat the Step 2 until 0 is obtained in the remainder. The last divisor which will be obtained will be the HCF.
Prime Factorisation Method
IN this method, every number must be treated as the product of prime factors. The product of the least common prime factor will give the HCF of those numbers.
HCF and LCM Formula = Product of two numbers = LCM of two numbers x HCF of two numbers.
Also Read: Average Value and Calculation
Questions and Answers on H.C.F and L.C.M
Solution: Let the numbers be 5m and 11m. Since 5:11 is already the reduced ratio, ‘m’ has to be the HCF. So, the numbers are 5 x 7 = 35 and 11 x 7 = 77.
Solution: Let us first convert each length to cm. So, the lengths are 450 cm, 990 cm, and 1620 cm. Now, we need to find the length of the largest plank that can be used to measure these lengths as the largest plank will take the least time. For this, we need to take the HCF of 450, 990, and 1620. 450 = 2 x 3 x 3 x 5 x 5 = 2 x 32 x 52 990 = 2 x 3 x 3 x 5 x 11 = 2 x 32 x 5 x 11 1620 = 2 x 2 x 3 x 3 x 3 x 3 x 5 = 22 x 34 x 5 Therefore, HCF (450, 990, 1620) = 2 x 3 x 3 x 5 = 90 Thus, we need a plank of length 90 cm to measure the given lengths in the least time.
Solution: The required number leaves remainders 1 and 4 on dividing 70 and 50 respectively. This means that the number exactly divides 69 and 46. So, we need to find the HCF of 69 (3 x 23) and 46 (2 x 23). HCF (69, 46) = 23 Thus, 23 is the required number.
Solution: To find the required number, we need to find the HCF of (136-64), (238-136), and (238-64), i.e., HCF (72, 102, 174). 72 = 23 x 32 102 = 2 x 3 x 17 174 = 2 x 3 x 29 Therefore, HCF (72, 102, 174) = 2 x 3 = 6 hence, 6 is the required number.
Solution: In these types of questions, we need to find the LCM of the divisors and add the common remainder (3) to it. So, LCM (5, 7, 9, 12) = 1260 Therefore, required number = 1260 + 3 = 1263
Solution: To find: LCM (6, 8).
The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, …
The multiples of 8 are 8, 16, 24, 32, 40, 48, ….
Thus, the smallest common multiple of 6 and 8 is 24.
Therefore, the LCM of 6 and 8 is 24.
Solution: To find the LCM of 4 and 12 using the prime factorisation method, follow the below steps.
Step 1: Find the prime factorization of given numbers:
The prime factorisation of 4 is 2 × 2
The prime factorisation of 12 is 2 × 2 × 3.
Step 2: The LCM of given numbers is found by multiplying the product of all factors. (Note: The common factor is included only once)
Hence, the product of prime factors = 2 × 2 × 3 = 12.
Therefore, the LCM of 4 and 12 is 12.
Solution: The prime factorisation of 54 is 2 × 3 × 3 × 3.
The prime factorisation of 60 is 2 × 2 × 3 × 5.
Thus, the product of prime factors = 2 × 2 × 3 × 3 × 3 × 5 = 540
Hence, the LCM of 54 and 60 is 540.
Solution: Given: LCM(12, 7) × HCF(12, 7) = Product(12, 7) …(1)
Finding LCM (12, 7):
The multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, ..
The multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, 120, …
Hence, the LCM of 12 and 7 is 84.
Finding HCF (12, 7):
The factors of 7 are 1 and 7.
The factors of 12 are 1, 2, 3, 4, 6 and 12.
Hence, HCF of 12 and 7 is 1.
Finding Product of 12 and 7:
The product of 12 and 7 = 12 × 7 = 84.
Now, substitute the obtained values in (1), we get
84 × 1 = 84
84 = 84
Hence, LHS = RHS
Therefore, LCM(12, 7) × HCF(12, 7) = Product(12, 7) is proved.
Solution: Given fractions: 2/5, 4/7 and 6/11
As we know, the formula to find the LCM of fractions is:
LCM of fractions = LCM of Numerators/HCF of Denominators .. (1)
Thus, the LCM of Numerators = LCM (2, 4, 6) = 12.
HCF of denominators = HCF (5, 7, 11) = 1
Now, substitute the values in (1), we get
LCM of fractions = 12/1 = 12
Hence, the LCM of the fractions 2/5, 4/7 and 6/11 is 12.
1. 3800
2. 4200
3. 4400
4. 3200
5. None of these
Solution: Option 2
25 = 5 × 5, 30 = 5 × 3 × 2, 35 = 5 × 7, 40 = 2 × 2 × 2 × 5
Required LCM = 2 × 2 × 2 × 5 × 5 × 3 × 7 = 4200
1. 193
2. 183
3. 223
4. 213
5. 233
Solution: Option 4
H.C.F. of 639 and 1065 is 213. H.C.F. of 213 and 1491 is 213.
1. 4/189
2. 6/63
3. 2/63
4. 20/21
5. None of these
Solution: Option 3
H.C.F of 4/9, 10/21 and 20/63 = H.C.F of 4,10 and 20 / L.C.M of 9,21 and 63
= H.C.F of 4, 10 and 20 = 2 & L.C.M. of 9, 21 and 63 = 63. Required H.C.F. = 2/63
1. 66, 77
2. 70, 84
3. 94, 108
4. 84, 96
5. 66, 106
Solution: Option 4
The difference of requisite numbers must be 12 and each should be divisible by 12. Checking the options given, only the fourth option satisfies.
1. 1,174
2. 74,100
3. 29, 154
4. 29, 145
5. None of these
Solution: Option 4
Let the numbers be 29a and 29b. Then, 29a + 29b= 174 or 29(a + b) = 174 or, a + b = 174/29 = 6. Values of co-primes (with sum 6) is(1, 5).
So, the possible pairs of numbers is (29 x 1, 29 x 5) i.e. 29 and 145, which will become the answer.
Solution: The largest four-digit number is 9999. Now, LCM (15, 21, 28) = 420 On dividing 9999 by 420, we get 339 as the remainder. Therefore, the required number is 9999-339 = 9660
Solution: They all will whistle again at the same time after an interval that is equal to the LCM of their individual whistle-blowing cycles. So, LCM (42, 60, 78) = 2 x 3 x 7 x 10 x 13 = 5460 Therefore, they will blow the whistle again simultaneously after 5460 sec, i.e., after 1 hour 31 minutes, i.e., at 11:01:00 hours.
Solution: LCM (6, 7, 8) = 168 So, the number is of the form 168m + 3. Now, 168m + 3 should be divisible by 9. We know that a number is divisible by 9 if the sum of its digits is a multiple of 9. For m = 1, the number is 168 + 3 = 171, the sum of whose digits is 9. Therefore, the required number is 171.
Solution: Let the common ratio be ‘m’. So, the numbers are 2m and 3m. Now, we know that the Product of numbers is = Product of LCM and HCF. => 2m x 3m = 294 => m2 = 49 => m = 7 Therefore, the numbers are 14 and 21.
Solution: We need to find the size of a square tile such that a number of tiles cover the field exactly, leaving no area unpaved. For this, we find the HCF of the length and breadth of the field. HCF (180, 105) = 15 Therefore, size of each tile = 15m x 15m Also, number of tiles = area of field / area of each tile => Number of tiles = (180 x 105) / (15 x 15) => Number of tiles = 84 Hence, we need 84 tiles, each of size 15m x 15m.
LCM and HCF Questions: PDF Link Available (Free Download)
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Tips for Preparing LCM and HCF Questions
Here are some tips to help you effectively prepare for LCM and HCF questions:
Understand the Basics:
- LCM is the smallest common multiple.
- HCF is the largest common factor.
Use Prime Factorization:
- Break numbers into their prime factors to find LCM and HCF easily.
Know the Formula:
- LCM × HCF = Product of the two numbers.
Practice Division and Listing:
- Use division or list multiples/factors for smaller numbers.
Solve Word Problems:
- Practice questions with real-life scenarios like time intervals or arrangements.
Try Different Types:
- Solve a mix of LCM and HCF problems for better understanding.
Use Shortcuts:
- Learn tricks to solve faster.
Review Past Papers:
- Practice questions from exams to understand patterns.
Draw Venn Diagrams:
- Use Venn diagrams to visualize common factors or multiples.
Practice Set on on H.C.F and L.C.M
1. What is the H.C.F of 18 and 24?
- A) 4
- B) 6
- C) 8
- D) 12
2. The L.C.M of 15 and 20 is:
- A) 40
- B) 50
- C) 60
- D) 75
3. Find the H.C.F of 36 and 48:
- A) 6
- B) 8
- C) 12
- D) 18
4. If the H.C.F of two numbers is 5 and their L.C.M is 60, what are the numbers?
- A) 5, 60
- B) 10, 30
- C) 15, 20
- D) 20, 30
5. The L.C.M of two co-prime numbers is always:
- A) A prime number
- B) A composite number
- C) An even number
- D) Odd number
6. Find the H.C.F of 56 and 72:
- A) 8
- B) 12
- C) 16
- D) 24
7. If the L.C.M of two numbers is 120 and their H.C.F is 5, find the product of the numbers:
- A) 100
- B) 120
- C) 150
- D) 200
8. What is the smallest number which leaves a remainder of 4 when divided by 6, 9, and 10?
- A) 64
- B) 94
- C) 124
- D) 154
9. The H.C.F of 72, 120, and 144 is:
- A) 12
- B) 18
- C) 24
- D) 36
10. If the L.C.M of two numbers is 45 and their H.C.F is 3, find the numbers:
- A) 15, 30
- B) 12, 36
- C) 9, 45
- D) 6, 54
11. The L.C.M of two numbers is 36, and one of the numbers is 12. Find the other number:
- A) 2
- B) 3
- C) 4
- D) 6
12. Find the H.C.F of 36, 54, and 72:
- A) 12
- B) 18
- C) 24
- D) 36
13. If the H.C.F of two numbers is 7 and their L.C.M is 105, what are the numbers?
- A) 7, 105
- B) 14, 15
- C) 21, 35
- D) 28, 30
14. The product of two numbers is 1200, and their H.C.F is 5. What is their L.C.M?
- A) 200
- B) 240
- C) 300
- D) 360
15. The L.C.M of two numbers is 80, and their H.C.F is 8. If one number is 16, find the other:
- A) 24
- B) 32
- C) 40
- D) 48
16. What is the H.C.F of 42 and 63?
- A) 7
- B) 14
- C) 21
- D) 28
17. The sum of two numbers is 80, and their H.C.F is 10. What is their L.C.M?
- A) 40
- B) 80
- C) 120
- D) 160
18. If the L.C.M of two numbers is 72 and their H.C.F is 9, find the numbers:
- A) 18, 36
- B) 24, 48
- C) 27, 45
- D) 36, 72
19. The H.C.F of three numbers is 3, and their L.C.M is 72. If two of the numbers are 9 and 12, find the third number:
- A) 3
- B) 4
- C) 6
- D) 8
20. If the H.C.F of two numbers is 9, and one of the numbers is 45, find the other number:
- A) 5
- B) 9
- C) 15
- D) 25
21. The L.C.M of three numbers is 120. If two of the numbers are 15 and 20, find the third number:
- A) 4
- B) 6
- C) 8
- D) 10
22. Find the H.C.F of 84, 126, and 168:
- A) 12
- B) 14
- C) 18
- D) 21
23. If the L.C.M of two numbers is 56 and their H.C.F is 8, find the numbers:
- A) 16, 24
- B) 20, 28
- C) 24, 32
- D) 28, 36
24. What is the H.C.F of 64 and 96?
- A) 16
- B) 24
- C) 32
- D) 48
25. The L.C.M of two numbers is 120, and their H.C.F is 6. If one number is 24, find the other:
- A) 36
- B) 48
- C) 60
- D) 72
26. Find the H.C.F of 56, 70, and 84:
- A) 14
- B) 18
- C) 21
- D) 28
27. If the H.C.F of two numbers is 4, and their L.C.M is 120, what are the numbers?
- A) 8, 30
- B) 12, 40
- C) 16, 60
- D) 24, 80
28. The sum of two numbers is 45, and their H.C.F is 9. What is their L.C.M?
- A) 45
- B) 81
- C) 90
- D) 135
29. If the L.C.M of two numbers is 72, and one of the numbers is 9, find the other number:
- A) 8
- B) 12
- C) 18
- D) 24
30. The H.C.F of 56, 98, and 126 is:
- A) 14
- B) 18
- C) 21
- D) 28
Answer Key for Practice Set
| 1.A) | 2.B) | 3.C) | 4.A) | 5.B) |
| 6.B) | 7.A) | 8.A) | 9.C) | 10.A) |
| 11.B) | 12.D) | 13.A) | 14.D) | 15.C) |
| 16.C) | 17.B) | 18.A) | 19.A) | 20.C) |
| 21.D) | 22.A) | 23.B) | 24.D) | 25.A) |
| 26.D) | 27.C) | 28.A) | 29.D) | 30.B) |
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