Very easy remainder

In other words, the problem means "Find the remainder when $6645^{6645}$ is divided by $6645*2$ . Therefore if $6645 \times x$ where $x$ is an even number, the remainder would be 0, else the remainder would be 6645.

Now, let's expand $6645^{6645}$ we get $6645 \times 6645 \dots \times 6645$ . Using number theory, we understand that odd times odd gives an odd number. If we simplify $6645^{6645}$ to $6645 \times x$ , we get that x is an odd number. Thus we get that the remainder is 6645, as concluded from the statement above

When dividing (x^x)/2x, the remainder is "0" when x is even and the remainder in "x" when x is odd. in the given problem x is 6645 which is odd. Hence the remainder is 6645.

take the example of 10 and 5 10 \5 rem = 5 and 10 /5^5 rem = 5 . so on the pattern we may conclude the answer to be 6645

We can eliminate the options 1 and 0 as 6645^6645 ends in 5 and the last digit of remainder on dividing by 13290 is 5..so the option left is 6645 ...that's our answer..