According to question ,

Number at (i) = 643 – 175 = 468
Number at (ii) = 1752 – 114 = 1638
Number at (iii) = 1149 – 213 = 936

Correct Option: C

According to question ,

Number at (i) = 643 – 175 = 468
Number at (ii) = 1752 – 114 = 1638
Number at (iii) = 1149 – 213 = 936
Clearly, 468, 1638 and 936 are multiples of 234 and 234 > 213.
∴  Divisor = 234

As we know that A number will be exactly divisible by 18 if it is divisible by 2 and 9 both. Clearly option ( d ) , 65043 is not divisible by 2.

Correct Option: D

As we know that A number will be exactly divisible by 18 if it is divisible by 2 and 9 both. Clearly option ( d ) , 65043 is not divisible by 2.
∴  Required number = 65043

Let the least number be x.

Correct Option: D

Let the least number be x.

y = 5 × 1 + 3 = 8
x = 13 × 8 + 1 = 105
On dividing 105 by 65,
⇒ 105 = ( 65 × 1 ) + 40
Hence required remainder = 40

As per the given above question ,
Here, 893 is exactly divisible by 47.
Hence, the required remainder is obtained on dividing 193 by 47.

Correct Option: B

As per the given above question ,
Here, 893 is exactly divisible by 47.
Hence, the required remainder is obtained on dividing 193 by 47.
⇒ 193 = ( 47 × 4 ) + 5
∴  Remainder = 5

According to question ,
The remainder will be same. On dividing 9 by 6, remainder = 3
∴ ( Number )² = 9² = 81

Correct Option: D

According to question ,
The remainder will be same. On dividing 9 by 6, remainder = 3
∴ ( Number )² = 9² = 81
On dividing 81 by 6
⇒ 81 = ( 6 × 13 ) + 3
Hence , Required remainder = 3