Top HCF and LCM Problems and Solutions

What is LCM and HCF?

HCF ( Highest Common Factor ) / GCF ( Greatest Common Factor ) HCF of given numbers is the greatest number which divides each number.
LCM: ( Lowest common multiple ). The least number which is divisible by the given numbers is called LCM.

HCF and LCM of given Fractions:

• HCF = HCF of Numerator / LCM of Denominator
• LCM = LCM of Numerator / HCF of Denominator
   

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Example : Find the HCF and LCM. of 0.63, 1.05 and 2.1. Sol. 
    
Solutions : Making the same number of decimal places,

the given numbers are 0.63, 1.05 and 2.10. 

Without decimal places, these numbers are 63, 105 and 210. 

Now, HCF of 63, 105 and 210 is 21. 

HCF of 0.63, 1.05 and 2.1 is 0.21. 

LCM of 63, 105 and 210 is 630. 

LCM of 0.63, 1.05 and 2.1 is 6.30. 

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Example 2: Two numbers are in the ratio of 15:11. If their H.C.F. is 13, find the numbers.

Solutions: Suppose the required numbers be 15x and llx,

Then, their HCF is x. So, x = 13. 

The numbers are (15 x 13 and 11 x 13)  ⇔ 195 and 143. 

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Example 3. Find the largest number which divides 62, 132 and 237 to leave the same remainder in each case.
 
Solutions : HCF of (132 - 62), (237 - 132) and (237 - 62)      

                  =  HCF of 70, 105 and 175 = 35

Hence The Required Number is 35 

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Example 4: Find the largest number of four digits exactly divisible by 12,15,18 and 27. 

Solutions: The Largest number of four digits is 9999.
     
Required number must be divisible by L.C.M. of 12,15,18,27 i.e. 540.
     
On dividing 9999 by 540,we get 279 as remainder .
 
Required number = (9999-279) = 9720.

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Example 5: Find the smallest number of five digits exactly divisible by 16, 24, 36 and 54. 

Solutions:  Five digits Smallest number is 10000.
     
Required number must be divisible by LCM of 16, 24, 36, 54 i.e 432,
       
Dividing 10000 by 432, we get 64 as remainder.
   
So Required number = 10000 +( 432 – 64 ) = 10368. 

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Example 6: Find the least number which when divided by 20,25,35 and 40 leaves remainders 14,19,29 and 34 respectively. 

Solutions: Here,(20-14) = 6,(25 – 19)=6,(35-29)=6 and (40-34)=6.
   
Required number = ( LCM. of 20, 25, 35, 40) – 6 =1394. 

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Example 7: Find the least number which when divided by 5,6,7, and 8 leaves a remainder 3, but when divided by 9 leaves no remainder. 

Solutions: LCM of 5, 6, 7, 8 = 840.
   
Required number is of the form ( 840k + 3 )
     
Least value of k for which (840k + 3) is divisible by 9 is k = 2.
 
Required number = (840 X 2 + 3)=1683

⇛ HCF and LCM Questions and Problems With Answer

1. 
   

   The H.C.F. of two numbers is 23 and the other two factors of their L.C.M. are 13 and 14. The larger of the two numbers is:

1. A. 276     
2. B. 299      
3. C. 322        Correct Answer
4. D. 345      

2.  

   
   

   Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10 and 12 seconds respectively. In 30 minutes, how many times do they toll together ?

1. A. 4     
2. B. 10      
3. C. 15      
4. D. 16        Correct Answer

3.  

   
   

   Let N be the greatest number that will divide 1305, 4665 and 6905, leaving the same remainder in each case. Then sum of the digits in N is:

1. A. 4       Correct Answer
2. B. 5      
3. C. 6      
4. D. 8      

4.  

   
   

   The greatest number of four digits which is divisible by 15, 25, 40 and 75 is:

1. A. 9000     
2. B. 9400      
3. C. 9600        Correct Answer
4. D. 9800      

5.  

   
   

   The product of two numbers is 4107. If the H.C.F. of these numbers is 37, then the greater number is:

1. A. 101     
2. B. 107      
3. C. 111        Correct Answer
4. D. 185      

6.  

   
   

   Three number are in the ratio of 3 : 4 : 5 and their L.C.M. is 2400. Their H.C.F. is:

1. A. 40       Correct Answer
2. B. 80      
3. C. 120      
4. D. 200      

7.  

   
   

   H.C.F. of 513, 1134 and 1215 is _____________

1. A. 18     
2. B. 27        Correct Answer
3. C. 33      
4. D. 36      

8.  

   
   

   The greatest number of four digits which is divisible by 15, 25, 40 and 75 is:

1. A. 9000     
2. B. 9400      
3. C. 9600        Correct Answer
4. D. 9800      

9.  

   
   

   Let N be the greatest number that will divide 1305, 4665 and 6905, leaving the same remainder in each case. Then sum of the digits in N is:

1. A. 4       Correct Answer
2. B. 5      
3. C. 6      
4. D. 8      

10.  

    
    

    Find the least number which is exactly divisible by 12, 15, and 20.

1. A. 40     
2. B. 50      
3. C. 60        Correct Answer
4. D. 80      

11.  

    
    

    Find the largest number of 4-digits divisible by 12, 15 and 18.

1. A. 9900       Correct Answer
2. B. 9750      
3. C. 9450      
4. D. 9000      

12.  

    
    

    The product of two numbers is 4107. If the H.C.F. of these numbers is 37, then the greater number is:

1. A. 101     
2. B. 107      
3. C. 111        Correct Answer
4. D. 185      

13.  

    
    

    Find H.C.F. of 0.63 and 1.05

1. A. 0.21       Correct Answer
2. B. 0.28      
3. C. 0.56      
4. D. 0.84      

14.  

    
    

    Find LCM. of 1.05 and 2.1

1. A. 1.3     
2. B. 1.25      
3. C. 2.1        Correct Answer
4. D. 4.30      

15.  

    
    

    Three number are in the ratio of 3 : 4 : 5 and their LCM. is 2400. Their H.C.F. is:

1. A. 40       Correct Answer
2. B. 80      
3. C. 120      
4. D. 200      

16.  

    
    

    H.C.F. of two numbers is 13. If these two numbers are in the ratio of 15: 11, then find the numbers.

1. A. 230, 140     
2. B. 215, 130      
3. C. 195, 143        Correct Answer
4. D. 155, 115      

17.  

    
    

    The ratio between two numbers is 4:5. If their LCM is 180, then the numbers are

1. A. 14, 24     
2. B. 16, 38      
3. C. 30, 42      
4. D. 36, 45        Correct Answer

18.  

    
    

    The H.C.F of two numbers is 5 and their L.C.M is 120. If one of the numbers is 40, find the other.

1. A. 15       Correct Answer
2. B. 21      
3. C. 23      
4. D. 28      

19.  

    
    

    The sum of two numbers is 495 and their HCF is 33. The numbers of such pairs is

1. A. 3     
2. B. 4        Correct Answer
3. C. 6      
4. D. 7      

20.  

    
    

    The LCM of two numbers is 12 times their H.C.F. The sum of the H.C.F and L.C.M is 403. If one number is 93, then the other is

1. A. 124       Correct Answer
2. B. 139      
3. C. 156      
4. D. 177
5.       

21.  

    
    

    The smallest number which is divisible by 12, 15, 25 and is a perfect square is

1. A. 3100     
2. B. 3350      
3. C. 3500      
4. D. 3600        Correct Answer

22.  

    
    

    128352/238368 when reduced to its lowest terms is

1. A. 5/11     
2. B. 7/13        Correct Answer
3. C. 9/17      
4. D. 11/13      

23.  

    
    

    The greatest number of 5 digits, which is divisible by 15, 20, 25 is

1. A. 99000     
2. B. 99900        Correct Answer
3. C. 99990      
4. D. 99999      

24.  

    
    

    If two numbers are in the ratio of 3:2 and the product of their HCF and LCM is 3750, then the sum of the numbers is

1. A. 125       Correct Answer
2. B. 133      
3. C. 147      
4. D. 155      

25.  

    
    

    The ratio between two numbers is 15:11. If their H.C.F is 13, then the numbers are

1. A. 141, 135     
2. B. 174, 145      
3. C. 195, 143        Correct Answer
4. D. 201, 157      

26.  

    
    

    What is the least number which when diminished by 4 is divisible by each one of 20, 30, 36 ?

1. A. 499     
2. B. 507      
3. C. 521      
4. D. 544        Correct Answer

27.  

    
    

    The HCF of three numbers is 10. If they are in the ratio 1 :2: 3, then the numbers are

1. A. 5, 10, 15     
2. B. 10, 20, 30        Correct Answer
3. C. 15, 30, 45      
4. D. 20, 40, 80      

28.  

    
    

    The H.C.F of two numbers is 25. The number which can be their L.C.M is

1. A. 80     
2. B. 95      
3. C. 116      
4. D. 125        Correct Answer

Find HCF and LCM based Problems




29.  

    
    

    Find the least number which when divided by 3, 4, 5, 6, 10 leaves the same remainder 3 in each case.

1. A. 63       Correct Answer
2. B. 65      
3. C. 69      
4. D. 72      

30.  

    
    

    What is the greatest number of 3 digits which when divided by 6, 9, 12 leaves a remainder 3 in each case ?

1. A. 850     
2. B. 925      
3. C. 975        Correct Answer
4. D. 1050      

31.  LCM HCF Problems below for further Reference

    
    

    
    

    
    

    The HCF and LCM of two numbers are 21 and 4641 respectively. If one of the number lies between 200 and 300, then the two numbers are

1. A. 251, 178     
2. B. 273, 357        Correct Answer
3. C. 323, 389      
4. D. 339, 421      

32.  

    
    

    Which greatest number will divide 3026 and 5053 leaving remainders 11 and 13 respectively ?

1. A. 45       Correct Answer
2. B. 51      
3. C. 57      
4. D. 63      

33.  

    
    

    The L.C.M of 3/4, 6/7, 9/8 is

1. A. 12     
2. B. 15      
3. C. 18        Correct Answer
4. D. 21      

34.  

    
    

    Find the HCF and LCM of 2/3, 8/9, 10/27 and 16/81

1. A. 3/70, 71/5     
2. B. 2/81, 80/3        Correct Answer
3. C. 5/83, 83/7      
4. D. 5/99, 92/7      

35.  

    
    

    The least perfect square number which is divisible by 4, 5, 6 and 8 is

1. A. 3200     
2. B. 3400      
3. C. 3600        Correct Answer
4. D. 3800      

36.  

    
    

    Two 3 digit numbers have their H.C.F 29 and L.C.M 4147. The sum of the numbers is

1. A. 635     
2. B. 696        Correct Answer
3. C. 733      
4. D. 749      

37.  

    
    

    The largest number that divides 210, 315 and 168 exactly is

1. A. 13     
2. B. 16      
3. C. 21        Correct Answer
4. D. 27      

38.  

    
    

    The LCM of two prime numbers a and b (a > b) is 161. The value of 2a - b is

1. A. 29     
2. B. 32      
3. C. 39        Correct Answer
4. D. 43      

39.  

    
    

    The HCF and LCM of two numbers are 10 and 60 respectively. If the sum of the numbers is 50, then one of the number will be

1. A. 5, 10     
2. B. 15, 20      
3. C. 20, 30        Correct Answer
4. D. 25, 35      

40.  

    
    

    If two numbers are in the ratio 7:5 and their L.C.M is 315, then their sum is

1. A. 108       Correct Answer
2. B. 123      
3. C. 135      
4. D. 141      

41.  

    
    

    The sum of two numbers is 50. Their difference is 1/5 of their sum. The L.C.M is

1. A. 54     
2. B. 55      
3. C. 57      
4. D. 60        Correct Answer

42.  

    
    

    A number which when divided by 10 leaves a remainder of 9, when divided by 9 leaves a remainder of 8, by 8 leaves a remainder of 7, etc., down to where, when divided by 2, it leaves a remainder of 1, is

1. A. 59     
2. B. 419      
3. C. 1259      
4. D. 2519        Correct Answer

43.  

    
    

    The greatest common divisor of 123456789 and 987654321 is

1. A. 1     
2. B. 3      
3. C. 9        Correct Answer
4. D. Greater than 9      

44.  

    
    

    What is the least number of cut pieces of equal length that can be cut out of two lengths 10 m 857 mm and 15 m 87 mm? 

1. A. 174     
2. B. 172      
3. C. 164      
4. D. 184        Correct Answer

45.  

    
    

    Philip, Tom and Brad start jogging around a circular field and complete a single round in 18 s, 22 s and 30 s, respectively. In how much time, will they meet again at the starting point?

1. A. 3 min 15 s     
2. B. 21 min      
3. C. 16 min 30 s        Correct Answer
4. D. 12 min