A number when successively divided by 5 and 6 gives remainders 3 and 2 respectively. What will be the remainders if the number is successively divided by 3 and 4 ?

Going in the reverse order 

When a number is divided by 6, the remainder is 2 => the number is of the form 6k+2

When 6k+2 is divided by 5,the remainder is 3 => the number is of the form 5(6k+2)+3 = 30k+13

When 30k+13 is divided by 3, the remainder is 1.

The remaining is 30k + 12, so when 30k+12 is divided by 4, the remainder is 2

This can also have another possibility of 1, 0 to be the remainders when successively divided by 3 and 4.

Among the given options 1, 2 satisfies this condition.