2023 AMC 12A Problem 4 — Solution | LIVE by Po-Shen Loh

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2023 AMC 12A Problem 4

Below is the professionally curated solution for Problem 4 of the 2023 AMC 12A, 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 of the real AMC 8, AMC 10, AMC 12, and AIME problems in our complete solution collection are used with official legal permission of the Mathematical Association of America (MAA).

Concepts:prime factorization digits

Difficulty rating: 1200

4.

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

$14$

$15$

$16$

$17$

$18$

Solution:

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

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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