= [(266 - 1)⁴⁰⁸¹ + 9]/266

= [( - 1)⁴⁰⁸¹ + 9]/266

= [- 1 + 9]/266

= 8/266

∴ Remainder = 8

Hence, option A is the correct answer.

For Ex: Consider the number 162

1 + 6 + 2 = 9, which is divisible by 3.

Hence, option B is the correct answer.

Divisibility of 9: A number is divisible by 9 if the sum of the digits of a number is divisible by 9.

5 + 7 + 2 + X + 4 + 1 = 19 + X

The next number to 19 which is divisible by 9 is 27.

∴ x = 27 – 19 = 8

Hence, option D is the correct answer.

= 105

Last three-digit number divisible by 15 = 990

Since, 990/15 = 66

Total 2-digit numbers divisible by 15 = 6 [15 × 6 = 90]

∴ Total three-digit numbers divisible by 15 = 66 – 6 = 60

Now, n = 60, First term, a = 105 and last term, l = 990

S_n = [n/2][a + l]

= [60/2][105 + 990]

= 32850

Hence, option C is the correct answer.

Let the digits at hundredth and tenth places be x and y respectively.

Divisibility of 11: A number if divisible by 11 if the difference of sum of digits at odd places and sum of digits at even places is either divisible by 11 or 0.

(4 + 6 + y) – (7 + x) = 10 + y – 7 – x

= 3 – x + y (i)

Divisibility of 3: A number is divisible by 3 if the sum of digits of a number is divisible by 3.

4 + 7 + 6 + x + y + 0

= 17 + x + y (ii)

By hit and trial

x = 8 and y = 5 satisfies both the equations.

Hence, option D is the correct answer.

Divisibility of 3: A number is divisible by 3 if the sum of the digits of a number is divisible by 3.

Now, 8 + 6 + y + 5 = 19 + y

The next number to 19 which is divisible by 3 is 21.

∴ y = 21 – 19 = 2

Hence, option D is the correct answer.

LCM of 16, 24, 72 and 84 = 1008

Greatest six-digit number = 999999

When we divide 999999 by 1008 it gives remainder 63.

∴ Required number = 999999 – 63 + 15

= 999951

Hence, option B is the correct answer.

Divisibility of 88 :- If a number is divisible by 8 and 11 both then the number is divisible by 88.

Divisibility of 8 :- If the last three digits of a number are divisible by 8, then the number is completely divisible by 8.

So, 8A4 is divisible by 8

For the least value A should be 2.

Divisibility of 11 :- A number is divisible by 11 if the difference between the sum of digits at its odd places and that of digits at the even places is either 0 or divisible by 11.

(2 + B + 8 + 4) – (7 + 5 + 2) = 0

14 + B – 14 = 0

Or, B = 14 – 14 = 0

∴ A + B = 2 + 0 = 2

Hence, option B is the correct answer.

b = 2a

c = 2b = 2 × 2a = 4a

d = 2c = 2 × 4a = 8a

Number = a × 1000 + 2a × 100 + 4a × 10 + 8a = 1248a

= 13 × 3 × 2⁵ × a

Hence, pin is divisible by 2, 3, 13.

Divisibility of 8 :- If the last three digits of a number are divisible by 8, then the number is completely divisible by 8.

89476*2 is divisible by 8

So, 6*2 is divisible by 8

632 is divisible by 8

*  = 3

"Hence, option A is the correct answer."