Find the number of digits in (875)^16.

Introduction / Context:We apply digit counting via base-10 logarithms. For N > 0, digits = floor(log10 N) + 1. Use factorization to compute log10 875 accurately.

Given Data / Assumptions:

• N = (875)^16.
• 875 = 7 × 125 = 7 × 5^3.

Concept / Approach:

• log10 875 = log10(7) + 3 log10(5).
• Then log10 N = 16 · log10 875; digits follow from flooring + 1.

Step-by-Step Solution:

Verification / Alternative check:(1000)^16 has 49 digits; since 875 < 1000, (875)^16 has fewer ⇒ 48 digits fits the estimate.

Why Other Options Are Wrong:

• 47 undercounts (would require N < 10^46).
• 49 or 50 overcount relative to (1000)^16.

Final Answer:48 digit