A number N is formed by writing 9 for 99 times. What is the remainder if N is divided by 13?

Correct Answer: (A) 11

A block of six 9s ($999,999$) is completely divisible by 13 ($999999 = 13$$\times$ 76923). The 99-digit number consisting of all 9s can be partitioned into 16 full blocks of six 9s ($16$$\times$ 6 = 96 digits) and 3 remaining 9s ($99 - 96 = 3$). The remainder of $N$ divided by 13 is therefore equal to the remainder of $999$ divided by 13: $999 = 13$$\times$ 76 + 11. Thus, the remainder is 11.