AMC10 2023 A

AMC10 2023 A · Q14

AMC10 2023 A · Q14. It mainly tests Probability (basic), Divisibility & factors.

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 $11$?

从前$100$个正整数中随机选一个数，然后从该数的正整数因数中随机选一个。选出的因数能被$11$整除的概率是多少？

(A) \frac{4}{100} \frac{4}{100}

(B) \frac{9}{200} \frac{9}{200}

(D) \frac{11}{200} \frac{11}{200}

(E) \frac{3}{50} \frac{3}{50}

Answer

Correct choice: (B)

正确答案：（B）

Solution

In order for the divisor chosen to be a multiple of $11$, the original number chosen must also be a multiple of $11$. Among the first $100$ positive integers, there are 9 multiples of 11; 11, 22, 33, 44, 55, 66, 77, 88, 99. 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 $\boxed{\textbf{(B)}~\frac{9}{200}}.$ $11 = 1, 11 \Rightarrow \frac{1}{2}\\ 22 = 2 \times 11: 1, 2, 11, 22 \Rightarrow \frac{1}{2}\\ 33 = 3 \times 11: 1, 3, 11, 33 \Rightarrow \frac{1}{2}\\ 44 = 2^2 \times 11: 1, 2, 4, 11, 22, 44 \Rightarrow \frac{1}{2}\\ 55 = 5 \times 11: 1, 5, 11, 55 \Rightarrow \frac{1}{2}\\ 66 = 2 \times 3 \times 11: 1, 2, 3, 6, 11, 22, 33, 66 \Rightarrow \frac{1}{2}\\ 77 = 7 \times 11: 1, 7, 11, 77 \Rightarrow \frac{1}{2}\\ 88 = 2^3 \times 11: 1, 2, 4, 8, 11, 22, 44, 88 \Rightarrow \frac{1}{2}\\ 99 = 3^2 \times 11: 1, 3, 9, 11, 33, 99 \Rightarrow \frac{1}{2}$

为了选出的因数是$11$的倍数，原始选出的数也必须是$11$的倍数。前$100$个正整数中有9个$11$的倍数：11, 22, 33, 44, 55, 66, 77, 88, 99。我们对这9个数分别计算选出$11$倍数因数的概率，每个均为$1/2$。首先选到这9个倍数的概率为$\frac{9}{100}$，因此总概率为$\frac{9}{100} \cdot \frac{1}{2} = \frac{9}{200}$，答案为$\boxed{\textbf{(B)}~\frac{9}{200}}$。 $11 = 1, 11 \Rightarrow \frac{1}{2}\\ 22 = 2 \times 11: 1, 2, 11, 22 \Rightarrow \frac{1}{2}\\ 33 = 3 \times 11: 1, 3, 11, 33 \Rightarrow \frac{1}{2}\\ 44 = 2^2 \times 11: 1, 2, 4, 11, 22, 44 \Rightarrow \frac{1}{2}\\ 55 = 5 \times 11: 1, 5, 11, 55 \Rightarrow \frac{1}{2}\\ 66 = 2 \times 3 \times 11: 1, 2, 3, 6, 11, 22, 33, 66 \Rightarrow \frac{1}{2}\\ 77 = 7 \times 11: 1, 7, 11, 77 \Rightarrow \frac{1}{2}\\ 88 = 2^3 \times 11: 1, 2, 4, 8, 11, 22, 44, 88 \Rightarrow \frac{1}{2}\\ 99 = 3^2 \times 11: 1, 3, 9, 11, 33, 99 \Rightarrow \frac{1}{2}$

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