2023 AMC 10A Problem 5

Below is the professionally curated solution for Problem 5 of the 2023 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2023 AMC 10A solutions, or check the answer key.

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Concepts:prime factorization digits

Difficulty rating: 1200

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

$14$

$15$

$16$

$17$

$18$

Solution:

Factor everything into primes. $8^5 \cdot 5^{10} \cdot 15^5 = 2^{15} \cdot 5^{10} \cdot 3^5 \cdot 5^5 = 2^{15} \cdot 5^{15} \cdot 3^5 = 10^{15} \cdot 243.$ That's $243$ followed by $15$ zeros, so it has $3 + 15 = 18$ digits. Thus, E is the correct answer.

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Problem 5 in Other Years

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