HCF and LCM – Railway Maths

Total Questions: 50

21. Rajesh has 180 litres of Oil A and 220 litres of Oil B. He fills a number of identical containers with the two types of oil in a manner that each container has only one type of oil, and all containers are completely filled. What can be the maximum volume (in litres) of each container that Rajesh uses, so that all the oil that Rajesh has, of both the types, can be poured into these containers? [Group D 08/09/2022 (Afternoon)]

Correct Answer: (b) 20
Solution:

H.C.F of 180 and 220 = 20
Hence, The maximum volume of container = 20 litres

22. The HCF of (X⁴ – Y⁴), (X⁸ – Y⁸) and (X² – Y²) is: [Group D 08/09/2022 (Evening)]

Correct Answer: (a) (X – Y)(X + Y)
Solution:

(x⁴ − y⁴), (x⁸ − y⁸) and (x² − y²)
⇒ x⁴ − y⁴ = (x² − y²)(x² + y²)
= (x + y)(x − y)(x² + y²)
⇒ x⁸ − y⁸ = (x⁴ − y⁴)(x⁴ + y⁴)
= (x² − y²)(x² + y²)(x⁴ + y⁴)
= (x + y)(x − y)(x² + y²) (x⁴ + y⁴)
⇒ x² − y² = (x + y)(x − y)
Hence, H.C.F = (x + y)(x − y)

23. The highest common factor of any two distinct prime numbers is: [Group D 09/09/2022 (Evening)]

Correct Answer: (d) 1
Solution:

The highest common factor of any two distinct prime numbers is always 1.

24. Which of the following is a pair of co-prime numbers? [Group D 12/09/2022 (Afternoon)]

Correct Answer: (a) 81, 16
Solution:

A pair of numbers whose HCF is 1 called the pair of co-prime numbers
HCF of 81 and 16 = 1
therefore, option (a) is the right answer

25. The greatest number which divides 1876, 12503 and 6877 leaving 1,3 and 2 respectively is: [Group D 15/09/2022 (Morning)]

Correct Answer: (b) 625
Solution:

1876 − 1 = 1875,
12503 − 3 = 12500 and 6877 − 2 = 6875
H.C.F of 1875, 12500 and 6875 = 625
Therefore the greatest no. which divides 1876, 12503 and 6877 leaving 1, 3 and 2 respectively is 625

26. A farmer plants three different types of plants in equal numbers in a garden. All plants of a type are planted to form a rectangle, in which no rectangle contains more than one type, and no plants of any type are left out. After all the plants were planted, the aisle with Plant A had 70 rows, the rectangle with Plant B had 28 rows, and the rectangle with Plant C had 42 rows. Find the minimum number of plants of each type planted by the farmer in the garden. [Group D 16/09/2022 (Afternoon)]

Correct Answer: (c) 420
Solution:

Minimum number of plants of each type = L.C.M of (28, 42, 70) = 420

27. Let R be the greatest number which when divides 41, 71 and 91 leaves the same remainder. Find the LCM of R and 45. [Group D 20/09/2022 (Morning)]

Correct Answer: (b) 90
Solution:

H.C.F of (91 − 71), (71 − 41), (91 − 41)
H.C.F of 20, 30, 50 is 10
Now, the L.C.M of 10 and 45 = 90

28. Consider two numbers whose LCM + HCF = 504, and LCM - HCF = 456. If one of these two numbers is 96, find the other number. [Group D 27/09/2022 (Afternoon)]

Correct Answer: (c) 120 
Solution:

LCM + HCF = 504 (i) &
LCM − HCF = 456 (ii)
Solve eq (i) and eq (ii) we get, LCM = 480 & HCF = 24
We know that LCM × HCF = 1st number × 2nd number
⇒ 480 × 24 = 96 × 2nd number
⇒ 2nd number = 120

29. The LCM of fractions is calculated as [Group D 28/09/2022 (Afternoon)]

Correct Answer: (b) 30
Solution:

30. Find the ratio between the LCM and the HCF of 5, 15, and 20. [Group D 30/09/2022 (Morning)]

Correct Answer: (c) 12 : 1
Solution:

LCM of 5, 15 and 20 = 60
HCF of 5, 15 and 20 = 5
Ratio of LCM and HCF = 60 : 5 = 12 : 1