Question 1

Three number are in the ratio of 3 : 4 : 5 and their L.C.M. is 2400. Their H.C.F. is:

Accepted Answer

Let the numbers be 3x, 4x and 5x. Then, their L.C.M. = 60x. So, 60x = 2400 or x = 40. The numbers are (3 x 40), (4 x 40) and (5 x 40). Hence, required H.C.F. = 40.

Question 2

The greatest number which on dividing 1657 and 2037 leaves remainders 6 and 5 respectively, is:

Accepted Answer

Required number = H.C.F. of (1657 - 6) and (2037 - 5) = H.C.F. of 1651 and 2032 = 127.

Question 3

The product of two numbers is 2028 and their H.C.F. is 13. The number of such pairs is:

Accepted Answer

Let the numbers 13a and 13b. Then, 13a x 13b = 2028 =>ab = 12. Now, the co-primes with product 12 are (1, 12) and (3, 4). [Note: Two integers a and b are said to be coprime or relatively prime if they have no common positive factor other than 1 or, equivalently, if their greatest common divisor is 1 ] So, the required numbers are (13 x 1, 13 x 12) and (13 x 3, 13 x 4). Clearly, there are 2 such pairs.

Question 4

A rectangular courtyard 3.78 meters long 5.25 meters wide is to be paved exactly with square tiles, all of the same size. what is the largest size of the tile which could be used for the purpose?

Accepted Answer

3.78 meters =378 cm = 2 × 3 × 3 × 3 × 7 5.25 meters=525 cm = 5 × 5 × 3 × 7 Hence common factors are 3 and 7 Hence LCM = 3 × 7 = 21 Hence largest size of square tiles that can be paved exactly with square tiles is 21 cm.

Question 5

The L.C.M of 22, 54, 108, 135 and 198 is

Accepted Answer

22 = 2 x 11 54 = 2 × 3 3 108 = 2 2 × 3 3 135 = 3 3 × 5 198 = 2 × 3 2 × 11 ∴ L . C . M = 2 2 × 3 3 × 5 × 11 = 5940

Question 6

The L.C.M of two numbers is 495 and their H.C.F is 5. If the sum of the numbers is 100, then their difference is

Accepted Answer

Let the numbers be x and (100-x). Then, x 100 - x = 5 * 495 => x 2 - 100 x + 2475 = 0 => (x-55) (x-45) = 0 => x = 55 or x = 45 The numbers are 45 and 55 Required difference = (55-45) = 10

Question 7

The sum of two numbers is 528 and their H.C.F is 33. The number of pairs of numbers satisfying the above condition is

Accepted Answer

Let the required numbers be 33a and 33b. Then 33a +33b= 528 => a+b = 16. Now, co-primes with sum 16 are (1,15) , (3,13) , (5,11) and (7,9). Therefore, Required numbers are ( 33 x 1, 33 x 15), (33 x 3, 33 x 13), (33 x 5, 33 x 11), (33 x 7, 33 x 9) The number of such pairs is 4

Question 8

Product of two co-prime numbers is 117. Their L.C.M should be

Accepted Answer

H.C.F of co-prime numbers is 1. So, L.C.M = 117/1 =117

Question 9

The G.C.D of 1.08, 0.36 and 0.9 is

Accepted Answer

Given numbers are 1.08 , 0.36 and 0.90 H.C.F of 108, 36 and 90 is 18 [ ∵ G.C.D is nothing but H.C.F] Therefore, H.C.F of given numbers = 0.18

HCF and LCM Questions