Find the number of digits in (875)^16.
Aptitude
Logarithm
Difficulty: Medium
Choose an option
-
A47 digit
-
B48 digit
-
C49 digit
-
D50 digit
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ENone of these
Answer
Correct Answer: 48 digit
Explanation
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:
log10 7 ≈ 0.845098, log10 5 ≈ 0.69897log10 875 ≈ 0.845098 + 3(0.69897) = 0.845098 + 2.09691 = 2.942008log10 N ≈ 16 × 2.942008 = 47.072128Digits = floor(47.072128) + 1 = 47 + 1 = 48Verification / 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