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 ?
 A 8:30:00 am B 9:00:00 pm C 12:00:00 pm D 12:00:00 am
Numbers    LCM
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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 co-prime numbers is 117. What is the sum of squares of the numbers ?
 A 220 B 1530 C 250 D 22
Numbers    LCM
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Question 2 Explanation:
117 = 3 x 3 x 13 As the numbers are co-prime, HCF = 1. So, the numbers have to be 9 and 13. 92 = 81 132 = 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 ?
 A 132 B 35 C 12 D 36
Numbers    LCM    HCF
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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 co-primes. => 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 ?
 A 216 B 210 C 72 D 840
Numbers    LCM    HCF
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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 = 212 - 152 = 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
 A (i) B (ii) C (iii) D (iv)
Numbers    LCM    Number Divisibility
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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?
 A 25 B 45 C 40 D 50
LCM
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Question 6 Explanation:
1st number = 3x
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?
 A 20 B 30 C 40 D 50
LCM
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Question 7 Explanation:
say, 1st number =3x
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:
 A 40 B 20 C 30 D 10
LCM
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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?
 A 120 B 180 C 360 D 240
LCM
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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?
 A 1683 B 1263 C 843 D 423
LCM
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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
 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?
 A 30 B 25 C 15 D 20
LCM    HCF
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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 ?
 A 1386 B 1246 C 1122 D 1432
LCM
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Question 12 Explanation:
LCM is 1386
 Question 13
What is the least number which is exactly divisible by 6, 12, 9 and 18?
 A 96 B 36 C 24 D 81
LCM
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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?
 A 26 minutes 18 seconds B 42 minutes 36 seconds C 45 minutes D 46 minutes 12 seconds
LCM
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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:
 A 1677 B 1683 C 2523 D 3363
LCM
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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.

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