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Harvard-MIT Math Tournament

Number theory Difficulty 4.3 AIME Find the answer United States

Problem:
If a positive integer multiple of $864$ is picked randomly, with each multiple having the same probability of being picked, what is the probability that it is divisible by $1944$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:
The probability that a multiple of $864 = 2^{5} 3^{3}$ is divisible by $1944 = 2^{3} 3^{5}$ is the same as the probability that a multiple of $2^{2}$ is divisible by $3^{2}$ , which since $4$ and $9$ are relatively prime is $\frac{1}{9}$ .

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