1975 AMC 12 Problem 18

Attempt Problem 18 of the 1975 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 1975 AMC 12 solutions, or check the answer key.

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

18.

A positive integer

N

with three digits in its base ten representation is chosen at random, with each three-digit number having an equal chance of being chosen. The probability that

\log_2N

is an integer is

0

\frac3{899}

\frac1{225}

\frac1{300}

\frac1{450}

Answer: D

Concepts:basic probability power of 2 counting integers in a range

Difficulty rating: 1340

Small hint:

An integral base-

2

logarithm means

N

is a power of

2

Big hint:

List the powers of

2

from

100

through

999

Solution:

There are

999-100+1=900

three-digit integers. The three powers of

2

among them are

128,256,

and

512.

Thus the probability is

\frac{3}{900}=\frac{1}{300}.

Therefore, the correct answer is D.