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2023 AMC 10A Problems/Problem 14: Difference between revisions

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Among the first <math>100</math> 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 <math>\frac{9}{100}</math>, so the final probability is <math>\frac{9}{100} \cdot \frac{1}{2} = \frac{9}{200}</math>, so the answer is <math>\boxed{B}.</math>
Among the first <math>100</math> 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 <math>\frac{9}{100}</math>, so the final probability is <math>\frac{9}{100} \cdot \frac{1}{2} = \frac{9}{200}</math>, so the answer is <math>\boxed{B}.</math>
Casework:
11: 11 - 1/2
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
~vaisri

Revision as of 15:39, 9 November 2023

A number is chosen at random from among the first $100$ 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 $11$?

$\textbf{(A)}~\frac{4}{100}\qquad\textbf{(B)}~\frac{9}{200} \qquad \textbf{(C)}~\frac{1}{20} \qquad\textbf{(D)}~\frac{11}{200}\qquad\textbf{(E)}~\frac{3}{50}$

Solution 1

Among the first $100$ 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 $\frac{9}{100}$, so the final probability is $\frac{9}{100} \cdot \frac{1}{2} = \frac{9}{200}$, so the answer is $\boxed{B}.$

Casework: 11: 11 - 1/2 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

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