1999 AMC 12 Problem 6

Attempt Problem 6 of the 1999 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 1999 AMC 12 solutions, or check the answer key.

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

What is the sum of the digits of the decimal form of the product

2^{1999} \cdot 5^{2001}?

2

4

5

7

10

Pair each factor of

2

with a factor of

5

to make

10

2^{1999} \cdot 5^{2001} = 5^2 \cdot 10^{1999}

Write

\begin{aligned} 2^{1999} \cdot 5^{2001} &= 2^{1999} \cdot 5^{1999} \cdot 5^2 \\ &= 25 \cdot 10^{1999}, \end{aligned}

which is

25

followed by

1999

zeros. The sum of the digits is

2 + 5 = 7.

Thus, the correct answer is D.

Problem 6 in other years