Explanation

The number of digits of N is ⌊log₁₀N⌋ + 1. Here log₁₀(125^100) = 100·log₁₀125 = 100·3·log₁₀5 = 300·log₁₀5. Since log₁₀5 = log₁₀(10/2) = 1 − log₁₀2 = 1 − 0.301 = 0.699, we get 300 × 0.699 = 209.7. Hence the number has ⌊209.7⌋ + 1 = 210 digits. Options (A)–(C) come from dropping the +1 or mishandling the factor 3, so the answer is (D) 210.