HCF and LCM – Railway Maths Part III

Total Questions: 52

41. Let N be the least number which when divided by 15, 25, 35, 40 and 42, the remainder in each case is 1 and N is divisible by 271. What is the sum of digits of N? [Level 4 (10/05/2022) Shift 1]

Correct Answer: (c) 13
Solution:

LCM of (15, 25, 35, 40, 42) is 840
So required number is (840k + 1) and it is also divisible by 271
27k + 1 is divisible by 271 for k = 10
So required number = 840 × 10 + 1 = 84001
Sum of the digits of number = (8 + 4 + 0 + 0 + 1) = 13

42. What is the smallest cubic number, which is divisible by 72, 108 and 300? [Level 5 (12/06/2022) Shift 1]

Correct Answer: (b) 27000
Solution:

72 = 2³ × 3²
108 = 2² × 3³
300 = 2² × 3 × 5²
LCM = 2³ × 3³ × 5²
So, 8 × 27 × 125 = 27000
27000 is the smallest cube number.

43. The LCM of two numbers is 20 times their HCF, and the sum of the LCM and the HCF is 504. If the difference of the numbers is 24. Then find the sum of the numbers. [Level 5 (12/06/2022) Shift 1]

Correct Answer: (d) 216
Solution:

Let the HCF = x
LCM = 20x
As per question,
x + 20x = 504 ⇒ x = 24
So, HCF = 24 and LCM = 480
Let the numbers = a and b
We know,
LCM × HCF = Product of the numbers
ab = 11520
(24)² = (a + b)² − 4 × 11,520
576 + 46,080 = (a + b)²
(a + b) = √46,656 = 216
So, sum of numbers = 216

44. What is the difference between the LCM and the HCF of 24 and 18? [Level 5 (12/06/2022) Shift 2]

Correct Answer: (d) 66
Solution:

LCM of 24 and 18 = 72
HCF of 24 and 18 = 6
Difference between LCM and HCF = 72 − 6 = 66

45. The LCM of two numbers is 35 times their HCF. The LCM and HCF differ by 238. If the two numbers differ by 14, then find the sum of the numbers. [Level 5 (12/06/2022) Shift 2]

Correct Answer: (a) 84
Solution:

Let the HCF = x
LCM = 35x
As per question,
35x − x = 238 ⇒ x = 7
HCF = 7
LCM = 35x = 35 × 7 = 245
We know,
LCM × HCF = Product of numbers
7 × 245 = ab ⇒ ab = 1715
a − b = 14 (given)
(a + b)² = (a − b)² + 4ab
(a + b)² = 14² + 4 × 1715
(a + b)² = 7056 ⇒ (a + b) = 84
So, sum of number = 84

46. The ratio of two numbers is 11 : 4 and their HCF is 16. What is the sum of these two numbers? [Level 2 (13/06/2022) Shift 1]

Correct Answer: (b) 240
Solution:

Ratio of two number = 11 : 4
HCF = 16
So, numbers are 11 × 16 = 176 and 4 × 16 = 64
Required sum of number = 176 + 64 = 240

47. A circus runs two shows, one with a lion and another with an elephant, separately. The show with the elephant runs for 80 minutes, and the show with the lion runs for 120 minutes. Both shows start at 6 p.m. At what time will both shows start again at the same time? [Level 2 (13/06/2022) Shift 1]

Correct Answer: (a) 10:00 p.m.
Solution:

LCM of 80 and 120 = 240 min
So, they will meet after 240 min or 4 hours
So, required time = 6 pm + 4 = 10 pm

48. The least perfect square which is divisible by each of 27, 54 and 72 is [Level 2 (13/06/2022) Shift 1]

Correct Answer: (a) 1296
Solution:

LCM of 27, 54 and 72 = 216
Number = 216k (possible value of k = 6)
So, number = 216 × 6 = 1296

49. x is the smallest six-digit number which, when divided by 20, 21, 22, 24 and 28, the remainder in each case is 5. What will be the quotient when x is divided by 462? [Level 2 (13/06/2022) Shift 2]

Correct Answer: (b) 220
Solution:

Smallest six digit number = 100000
20 = 2² × 5
21 = 3 × 7
22 = 2 × 11
24 = 2³ × 3
28 = 2² × 7
LCM of 20, 21, 22, 24 and 28
= 2³ × 5 × 7 × 11 × 3 = 9240
Number = 9240k + 5 = 101645
When 101645 is divided by 462,
Quotient = 220

50. When 6892, 7105 and 7531 are divided by the greatest number x, then the remainder in each case is the same. The sum of the digit of x is: [Level 3 (14/06/2022) Shift 1]

Correct Answer: (d) 6
Solution:

7105 − 6892 = 213
7531 − 7105 = 426
HCF of 6892, 7105, and 7531 = 213
Sum of digits = 2 + 1 + 3 = 6