LCM and HCF
Factors and Multiples : All the numbers that divide a number completely, i.e., without leaving any remainder, are called factors of that number. For example, 24 is completely divisible by 1, 2, 3, 4, 6, 8, 12, 24. Each of these numbers is called a factor of 24 and 24 is called a multiple of each of these numbers … More on LCM and HCF
Question 1 
Three friends started running together on a circular track at 8:00:00 am. Time taken by them to complete one round of the track is 15 min, 20 min, 30 min respectively. If they run continuously without any halts, then at what time will they meet again at the starting point for the fourth time ?
8:30:00 am  
9:00:00 pm  
12:00:00 pm  
12:00:00 am 
Discuss it
Question 1 Explanation:
LCM (15, 20, 30) = 60
=> They meet at the starting point after every 60 min, i.e., after every 1 hour.
Therefore, they will meet at the starting point for the fourth time after 4 hours, i.e., at 12:00:00 pm.
Question 2 
The LCM of two coprime numbers is 117. What is the sum of squares of the numbers ?
220  
1530  
250  
22 
Discuss it
Question 2 Explanation:
117 = 3 x 3 x 13
As the numbers are coprime, HCF = 1.
So, the numbers have to be 9 and 13.
9^{2} = 81
13^{2} = 169
Therefore, required answer = 250
Question 3 
HCF of two numbers is 11 and their LCM is 385. If the numbers do not differ by more than 50, what is the sum of the two numbers ?
132  
35  
12  
36 
Discuss it
Question 3 Explanation:
Product of numbers = LCM x HCF = 11 x 385 = 4235
Let the numbers be of the form 11m and 11n, such that 'm' and 'n' are coprimes.
=> 11m x 11n = 4235
=> m x n = 35
=> (m,n) can be either of (1, 35), (35, 1), (5, 7), (7, 5).
=> The numbers can be (11, 385), (385, 11), (55, 77), (77, 55).
But it is given that the numbers cannot differ by more than 50.
Hence, the numbers are 55 and 77.
Therefore, sum of the two numbers = 55 + 77 = 132
Question 4 
Two numbers are in the ratio of 5:7. If their LCM is 105, what is the difference between their squares ?
216  
210  
72  
840 
Discuss it
Question 4 Explanation:
Let 'h' be the HCF of the two numbers.
=> The numbers are 5h and 7h.
We know that Product of Numbers = LCM x HCF
=> 5h x 7h = 105 x h
=> h = 3
So, the numbers are 15 and 21.
Therefore, difference of their squares = 21^{2}  15^{2} = 441  225 = 216
Question 5 
Which of the following is the largest of all ?
(i) 7/8
(ii) 15/16
(iii) 23/24
(iv) 31/32
(i)  
(ii)  
(iii)  
(iv) 
Discuss it
Question 5 Explanation:
LCM (8, 16, 24, 32) = 96
7/8 = 84/96
15/16 = 90/96
23/24 = 92/96
31/32 = 93/96
Hence, 31/32 is the largest of all.
Question 6 
Two numbers are in the ratio 3 : 5. If their L.C.M. is 75. what is sum of the numbers?
25  
45  
40  
50 
Discuss it
Question 6 Explanation:
1st number = 3x
2nd number =5x
LCM of 3x and 5x is 15x
=> 15x = 75
=> x = 5
sum = 15+25 =40
2nd number =5x
LCM of 3x and 5x is 15x
=> 15x = 75
=> x = 5
sum = 15+25 =40
Question 7 
Ratio of two numbers is 3:2. If LCM of numbers is 60, then smaller number is?
20  
30  
40  
50 
Discuss it
Question 7 Explanation:
say, 1st number =3x
2nd number =2x
LCM of numbers = 6x
given LCM = 60
=> x6 = 60
=>x = 10
2nd number =2x
LCM of numbers = 6x
given LCM = 60
=> x6 = 60
=>x = 10
Question 8 
Three numbers are in the ratio of 2 : 3 : 4 and their L.C.M. is 240. Their H.C.F. is:
40  
20  
30  
10 
Discuss it
Question 8 Explanation:
Let the numbers be 2x, 3x and 4x
LCM = 12x
12x=240
⇒x=20
H.C.F of 40, 60 and 80=20
Question 9 
What is the lowest common multiple of 12, 36 and 20?
120  
180  
360  
240 
Discuss it
Question 9 Explanation:
LCM = 180
Question 10 
What is the least number which when divided by 4, 5, 6 and 7 leaves a remainder 3, but when divided by 9 leaves no remainder?
1683  
1263  
843  
423 
Discuss it
Question 10 Explanation:
LCM of 4,5,6,7 is 420
=>(420k+3) should be divisible by 9
if k =1, 423/9 remainder !=0
=>(420k+3) should be divisible by 9
if k =1, 423/9 remainder !=0
Question 11 
The LCM. of two numbers is 30 and their HCF. is 15. If one of the numbers is 30, then what is the other number?
30  
25  
15  
20 
Discuss it
Question 11 Explanation:
Say another number =x
product of two numbers = product of HCF and LCM
x.30 = 15*30
x=15
Question 12 
What is the least number which is exactly divisible by 6, 7, 9 and 11 ?
1386  
1246  
1122  
1432 
Discuss it
Question 12 Explanation:
LCM is 1386
Question 13 
What is the least number which is exactly divisible by 6, 12, 9 and 18?
96  
36  
24  
81 
Discuss it
Question 13 Explanation:
LCM is 36
Question 14 
A, B and C start at the same time in the same direction to run around a circular track. A completes a round in 252 seconds, B in 308 seconds and C in 198 seconds, all starting at the same point. When will they meet again at the starting point?
26 minutes 18 seconds
 
42 minutes 36 seconds
 
45 minutes
 
46 minutes 12 seconds

Discuss it
Question 14 Explanation:
LCM of 252, 308 and 198 = 2772. Therefore, they will meet at the starting point again after 2772 seconds = 46 minutes and 12 seconds.
Question 15 
The least number which when divided by 5, 6, 7 and 8 leaves a remainder of 3, but when divided by 9 leaves no remainder, is:
1677
 
1683
 
2523
 
3363

Discuss it
Question 15 Explanation:
LCM of 5, 6, 7 and 8 = 840.
Therefore, required number is of the form 840y + 3.
Least value of y for which 840y+3 is divisible by 9 is 2.
For y=2, 840y+3 = 1683.
There are 15 questions to complete.