2023 AMC 10A Problem 5

Problem 5 of 25EasierNumber Theory

How many digits are in the base-ten representation of

8^5 \cdot 5^{10} \cdot 15^5?

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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.

Concepts: prime factorization · digits

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.