We know that,

LCM × HCF = Product of the numbers

Or, 126 × 9 = 18 × other number

Or, other number =  = 63

Hence, option A is the correct answer.

Let the numbers be 2x,3x,5x and 7x

HCF of 2x,3x,5x and 7x = x

x = 6

largest number = 7x = 7×6 = 42

Smallest number = 2x = 2×6 = 12

Sum of the squares of the largest and the smallest numbers = 42² + 12² = 1764 + 144 = 1908

18 = 2 × 3²

21 = 3 × 7

24 = 2³ × 3

LCM of 18,21 and 24 = 2³ × 3² × 7 = 504

Five digits smallest number = 10000

When 10000 is divide by 504 remainder is 424 (504 × 19 + 424)

Required number = 10000 + (504 – 424) = 10080

15 = 3×5

24 = 2³×3

28 =2²×7

32 =2⁵

LCM of 15, 24, 28 and 32 =2⁵×3×5×7 = 3360

According to the option,

Required number = 3360 × 3 + 7 =10087

Let the HCF of two numbers be x

LCM of two numbers = 56x

Given:

 x + 56x = 1710

 57x = 1710

 x = 30

HCF of the two numbers = 30

LCM of the two numbers = 56x = 56 × 30 = 1680

We know that:

First number × second number = LCM × HCF

 240 × second number = 1680 × 30

 Second number =

 Second number = 210

Therefore, the other number is 210

Hence, option D is correct.

Let the two numbers be 9x and 13x

Given, HCF of 9x and 13x = x = 6

First number = 9x = 9 × 6 = 54

Second number = 13x = 13 × 6 = 78

We know that:

First number × second number = LCM × HCF

⇒ 54 × 78 = LCM × 6

⇒ LCM =

⇒ LCM = 702

Therefore, the LCM of both numbers is 702.

Hence, option C is correct.

The greatest number by which when 448, 678 and 908 are divided, the remainder in each case is 11 = HCF of (448 - 11), (678 - 11) and (908 - 11)

x = HCF of (437, 667 and 897)

x = 23

Therefore, required remainder when 147 is divided by 23 =  = 9 remainder

Hence, option C is correct.

Let one number be x

Other number = 140 – x

We know that:

First number × second number = LCM × HCF

 (x) × (140 - x) = 240 × 20

 140x – x² = 4800

 x² – 140x + 4800 = 0

 x² – 80x – 60x + 4800 = 0

 x(x - 80) – 60(x - 80) = 0

 (x - 60)(x - 80) = 0

 x = 60, 80

If, x = 60

Other number = 140 – 60 = 80

If, x = 80

Other number = 140 – 80 = 60

Therefore, the smaller number = 60

Hence, option A is correct.

We know that:

First number × second number = LCM × HCF

48 × 144 = LCM × 48

⇒ LCM = 144

Hence, option C is correct.