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

Factor each base into primes:

8=2^3,

15=3\cdot 5

Pair every

2

with a

5

to form a power of

10,

leaving a small factor

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.

Problem 4 in other years