1984 AMC 12 Problem 9

Attempt Problem 9 of the 1984 AMC 12 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 1984 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

The number of digits in

4^{16}5^{25}

(when written in the usual base

10

form) is

31

30

29

28

27

Rewrite the power of

4

as a power of

2

Pair

25

factors of

2

with the

25

factors of

5

We have

\begin{aligned} 4^{16}5^{25} &=2^{32}5^{25}\\ &=2^7\cdot10^{25}\\ &=128\cdot10^{25}. \end{aligned}

This is

128

followed by

25

zeros, so it has

3+25=28

digits. Therefore, the correct answer is D.

Problem 9 in other years