1981 AMC 12 Problem 16

Attempt Problem 16 of the 1981 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 1981 AMC 12 solutions, or check the answer key.

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

16.

The base three representation of

x

12112211122211112222.

The first digit (on the left) of the base nine representation of

x

1

2

3

4

5

Since

9=3^2,

group the base-three digits in pairs from the right

Convert the leftmost pair

12_3

into a base-nine digit

The representation has an even number of base-three digits, so its leftmost base-nine digit comes from

12_3.

That pair has value

1\cdot3+2=5,

so the first base-nine digit is

5.

Therefore, the correct answer is E.

Problem 16 in other years