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
, so the answer is
\\
\\
22 = 2 * 11: 11, 22 - 1/2\\
33 = 3 * 11: 11, 33 - 1/2\\
44 = 2^2 * 11: 11, 22, 44 - 1/2\\
55 = 5 * 11: 11, 55 - 1/2\\
66 = 2 * 3 * 11: 11, 22, 33, 66 - 1/2\\
77 = 7 * 11: 11, 77 - 1/2\\
88 = 2^3 * 11: 11, 22, 44, 88 - 1/2\\
99 = 3^2 * 11: 11, 33, 99 - 1/2$
~vaisri