2023 AMC 10A Problem 5

Attempt Problem 5 of the 2023 AMC 10A 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 10A 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:

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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