Explanation

N ≡ 1 (mod 2ⁿ), so N = 2ⁿk + 1. Taking k = 1, the smallest 5-digit number of this form is 2ⁿ + 1 ≥ 10000, i.e. 2ⁿ ≥ 9999. Since 2¹³ = 8192 < 9999 < 16384 = 2¹⁴, the value is n = 14 (giving N = 16385), option (c).