N is positive integer and hundredth digit of 10N is 6

This question should be re-framed by specifying that N is a two digit positive integer.

1) If the hundred's digit of 10N is 6 => the ten's digit of N is 6.
2) The ten's digit of N+13 is 7 => the one's digit of N is 5

The one's digit of N is 5 because for the ten's digit of N to be 6, and the ten's digit of N+13 to be 7, the addition of 13 to N must be causing a carry of 1 to the ten's column. Therefore 3 must be getting added to the ten's column. Now, if the one's digit is X, we must choose X in such a way that 60+X and 70+(X+3) are both divisible by 13. This is possible when X=5.

Option (B), provided that this number is two digit number (not given in the question).

If this is not a two digit number, then the numbers 260 and 273 will also satisfy this criteria and in that case the units digit will be 0, 364 and 377 will also satisfy with the units digit being 4, 663 and 676 will also satisfy with the units digit being 3....... so there is no unique answer for the units digit in that case.