Correct Option (A)

: 11

To determine the remainder when N (a number formed by writing 9 for 99 times) is divided by 13, we analyze the pattern of remainders for repeating sequences of the digit 9 when divided by 13.

9 divided by 13 yields a remainder of 9.
99 divided by 13 yields a remainder of 8.
999 divided by 13 yields a remainder of 11.
9999 divided by 13 yields a remainder of 2.
99999 divided by 13 yields a remainder of 3.
999999 divided by 13 yields a remainder of 0.

This establishes that a number consisting of six consecutive 9s (999999) is perfectly divisible by 13. Thus, the cycle length for the remainder is 6.

The number N consists of 99 nines. To find the effective number of nines contributing to the remainder, we divide 99 by the cycle length:

99 ÷ 6 = 16 with a remainder of 3.

This indicates that N can be conceptualized as 16 blocks of '999999' followed by three 9s ('999'). Since each block of '999999' has a remainder of 0 when divided by 13, the remainder of N will be solely determined by the remainder of the last three 9s (999) when divided by 13.

From the established pattern, 999 divided by 13 yields a remainder of 11.

Therefore, the remainder when N is divided by 13 is 11.

Incorrect Options:

Options (B) 9, (C) 7, and (D) 1 are incorrect because the systematic calculation of remainders based on the cyclic property of repeating digits when divided by 13 conclusively yields 11 as the remainder.