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

Factor each base into primes:

8 = 2^3

and

15 = 3 \cdot 5

Combine into

10^{15} \cdot 3^5,

which is

3^5

followed by

15

zeros

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.

Problem 5 in other years