2023 AMC 10A Problems/Problem 14

A number is chosen at random from among the first positive integers, and a positive integer divisor of that number is then chosen at random. What is the probability that the chosen divisor is divisible by ?

Solution 1

Among the first positive integers, there are 9 multiples of 11. We can now perform a little casework on the probability of choosing a divisor which is a multiple of 11 for each of these 9, and see that the probability is 1/2 for each. The probability of choosing these 9 multiples in the first place is , so the final probability is $\frac{9}{100} \cdot \frac{1{2} = \frac{9}{200}$ (Error compiling LaTeX. Unknown error_msg), so the answer is