Solved Examples

1. LCM of 27,314and53is${\displaystyle \frac{2}{7},{\displaystyle \frac{3}{14}\mathrm{a}\mathrm{n}\mathrm{d}{\displaystyle \frac{5}{3}\mathrm{i}\mathrm{s}}}}$

a. 45

b. 35

c. 30

d. 25

Correct Option: C

Explanation:

LCM of numeratorsHCF of denominators=LCM of 2,3,5HCF of 7,14,3=301=30${\displaystyle \frac{\text{LCM of numerators}}{\text{HCF of denominators}}=\frac{\text{LCM of 2,3,5}}{\text{HCF of 7,14,3}}=\frac{30}{1}=30}$

2. About the number of pairs which have 16 as their HCF and 136 as their LCM, the conclusion can be

a. only one such pair exists

b. only two such pairs exist

c. many such pairs exist

d. no such pair exists

Correct Option: D

Explanation:

HCF is always a factor of LCM. ie., HCF always divides LCM perfectly.

3. The HCF of two numbers is 12 and their difference is also 12. The numbers are

a. 66, 78

b. 94, 106

c. 70, 82

d. 84, 96

Correct Option: D

Explanation:

The difference of required numbers must be 12 and every number must be divisible by 12. Therefore, they are 84, 96.

4. The HCF of two numbers is 16 and their LCM is 160.  If one of the numbers is 32, then the other number is 

a. 48

b. 80

c. 96

d. 112

Correct Option:b

Explanation:

The number = HCF×LCMGiven number=16×16032=80

5. HCF of three numbers is 12. If they are in the ratio 1:2:3, then the numbers are

a. 12,24,36

b. 10,20,30

c. 5,10,15

d. 4,8,12

Correct Option: A

Explanation:

Let the numbers be a, 2a and 3a.

Then, their HCF = a  so a=12

The numbers are 12,24,36

6. 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?

a. 4

b. 10

c. 15

d. 16

Correct Option: D

Explanation:

LCM of 2,4,6, 8,10 and 12 is 120. So, the bells will toll simultaneously after 120 seconds.

i.e.2 minutes. In 30 minutes, they (302+1)$\left({\displaystyle \frac{30}{2}+1}\right)$ toll times ie.16 times.

7. The largest natural number which exactly divides the product of any four consecutive natural numbers is :

a. 6

b. 12

c. 24

d. 120

Correct Option: C

Explanation:

The required number can be find out by following way.

1×2×3×4=24$1\times 2\times 3\times 4=24$

8. LCM of 87 and 145 is :
a. 870
b. 1305
c. 435
d. 1740
Correct Option: C
Explanation:
87 = 29 x 3
145 = 29 x 5
So LCM = 29 x 3 x 5 = 435

9. The traffic lights at three different road crossing change after every 48 sec; 72 sec; and 108 sec., respectively.  If they all change simultaneously at 8:20:00 hrs, then they will again change simultaneously at

a. 8:27:12 Hrs

b. 8:27:24 Hrs 

c. 8:27:36 Hrs

d. 8:27:48 Hrs

Correct Option: A

Explanation:

The change of interval=(LCM of 48,72,108)sec.=432. So, for every 432 seconds i.e.7 min. 12 sec. the lights will change.  So add 7 min.12 sec.to 8:20:00 Hrs.i.e.8:27:12 Hrs.

10. The greatest number by which if 1657 and 2037 are divided the remainders will be 6 and 5 respectively is

a. 127

b. 235

c. 260

d. 305

Correct Option: A

Explanation:

The needed number is HCF of (1657-6) and (2037-5)=HCF of 1651 & 2032=127.