Two distinct numbers a and b are chosen randomly from the set \2, 2^2, 2^3, …, 2^25\. What is the probability that \log_a b is an integer?

2005 AMC 12A Problem 23 — Solution

2005 AMC 12A Problem 23

Below is the professionally curated solution for Problem 23 of the 2005 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2005 AMC 12A solutions, or check the answer key.

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Concepts:divisibility floor and ceiling functions basic probability

Difficulty rating: 2330

23.

Two distinct numbers $a$ and $b$ are chosen randomly from the set $\{2, 2^2, 2^3, \ldots, 2^{25}\}.$ What is the probability that $\log_a b$ is an integer?

$\dfrac{2}{25}$

$\dfrac{31}{300}$

$\dfrac{13}{100}$

$\dfrac{7}{50}$

$\dfrac{1}{2}$

Solution:

Let $a = 2^j$ and $b = 2^k.$ Then $\log_a b = \dfrac{k}{j},$ which is an integer exactly when $j \mid k.$

For each $j,$ the number of valid $k \ne j$ in $\{1, \ldots, 25\}$ is $\left\lfloor \tfrac{25}{j} \right\rfloor - 1.$ Summing over $j$ gives $24 + 11 + 7 + 5 + 4 + 3 + 2 + 2 + 4 \cdot 1 = 62$ ordered pairs $(a, b).$

Since there are $25 \cdot 24 = 600$ ordered pairs of distinct elements, the probability is $\dfrac{62}{600} = \dfrac{31}{300}.$

Thus, the correct answer is B.

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