119 is divisible by 7, 187 is divisible by 11,247 is divisible by 13 and 551 is divisible by 19. So none of the given numbers is prime.

Correct Option: D

119 is divisible by 7, 187 is divisible by 11,247 is divisible by 13 and 551 is divisible by 19. So none of the given numbers is prime.

For even values of n, the number (10ⁿ - 1 ) consists of even numbers of nines and hence it will be divisible by 11.

Correct Option: C

For even values of n, the number (10ⁿ - 1 ) consists of even numbers of nines and hence it will be divisible by 11.

On dividing 1056 by 23, we get 21 as remainder
∴ Required number to be added = (23 - 21 ) = 2.

Correct Option: D

On dividing 1056 by 23, we get 21 as remainder
∴ Required number to be added = (23 - 21 ) = 2.

Let d = divisor and q = quotient

First remove the last remainder:
d x q + 413 = 380606
d x q = 380193

Correct Option: B

Let d = divisor and q = quotient

First remove the last remainder:
d x q + 413 = 380606
d x q = 380193

So d cannot be even (choice d) or a multiple of 5 (choice c).

If we try choice a or b,
a quick division (by calculator) shows that
380193 = 451 x 843, the other two choices.

(In the absence of choices, we could find the prime factors of 380193,
which are 3 x 11 x 41 x 281 and get various candidate divisors from that.)

So one of them is the divisor and the other the quotient.

So we can just check the first remainder:
843 x 4 = 3372
3806 - 3372 = 434 ... a match!

And just to be sure:
451 x 8 = 3608
3806 - 3608 = 198

So the divisor is 843.

6 + 1 + 3 + 5 + x + 2 = 17 + x must be divisible by 9.
So, x =1.

Correct Option: B

6 + 1 + 3 + 5 + x + 2 = 17 + x must be divisible by 9.
So, x =1.