**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

## LCM

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 |

**Numbers**

**LCM**

**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 co-prime numbers is 117. What is the sum of squares of the numbers ?

220 | |

1530 | |

250 | |

22 |

**Numbers**

**LCM**

**Discuss it**

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.
9

^{2}= 81 13^{2}= 169 Therefore, required answer = 250Question 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 |

**Numbers**

**LCM**

**HCF**

**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 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 ?

216 | |

210 | |

72 | |

840 |

**Numbers**

**LCM**

**HCF**

**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 = 216Question 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) |

**Numbers**

**LCM**

**Number Divisibility**

**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 |

**LCM**

**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 |

**LCM**

**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 |

**LCM**

**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 |

**LCM**

**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 |

**LCM**

**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

There are 15 questions to complete.