Let N be the greatest number that will divide 1305, 4665 and 6905 leaving the same remainder in each case. The sum of digits of N is
$
  (a){\text{ 4}} \\
  (b){\text{ 5}} \\
  (c){\text{ 6}} \\
  (d){\text{ 8}} \\
 $

Hint – Use the concept that if we have 3 numbers a, b and c then the greatest number that will divide all three of them leaving the same remainder will be the H.C.F of (b-a), (c-b), (c-a). Find the height common factor by writing down all the factors of each number obtained.

Complete step-by-step answer:

The given numbers are 1305, 4665 and 6905.

We have to find the greatest number N that will divide 1305, 4665 and 6905 leaving the same remainder.

The greatest number N is the H.C.F of (4665 – 1305), (6905 – 4665) and (6905 – 1305).
So the greatest number N is the H.C.F of 3360, 2240 and 5600 leaving the same remainder.

So first find out the factors of 3360, 2240 and 5600.

So factors of 3360 are
$ \Rightarrow 3360 = 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 7 \times 5$

Now the factors of 2240 are
$ \Rightarrow 2240 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 7 \times 5$

Now the factors of 5600 are
$ \Rightarrow 5600 = 2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 7$

Now as we know that the H.C.F of the numbers is the common factors of the numbers so the required H.C.F of the numbers are
$ \Rightarrow H.C.F = 2 \times 2 \times 2 \times 2 \times 2 \times 7 \times 5 = 1120$
Therefore N = 1120.

Now we have to calculate the sum of the digits of N.

$ \Rightarrow N = 1 + 1 + 2 + 0 = 4$
So 4 is the required answer.

Hence option (A) is correct.

Note – The above concept mentioned has a tricky part that it is applicable if c > b >a, if it’s not the case then eventually numbers obtained will be negative. But the magnitude can be preferred in those cases as eventually when H.C.F is taken for the number, by removing the negative sign will too give the right answer.