2023 AMC 12A Problem 4

Attempt Problem 4 of the 2023 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2023 AMC 12A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

How many digits are in the base-ten representation of $8^5\cdot 5^{10}\cdot 15^5?$

$14$

$15$

$16$

$17$

$18$

Answer: E

Concepts:prime factorization digits

Difficulty rating: 1200

Solution:

Writing everything in primes, $\begin{gathered} 8^5\cdot 5^{10}\cdot 15^5\\ {}=2^{15}\cdot 5^{10}\cdot 3^5\cdot 5^5\\ {}=3^5\cdot 2^{15}\cdot 5^{15}. \end{gathered}$

This equals $3^5\cdot 10^{15}=243\cdot 10^{15},$ which is $243$ followed by $15$ zeros, for a total of $18$ digits.

Thus, the correct answer is E.

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