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Multiples of 864

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

This can be easily solve by dividing out the greatest common divisor between 864 and 1944.

(1)   \begin{align*} (1944,864) &= (864,1080) \\ &= (864,216) \\ &= (216,648) \\ &= (216,432) \\ &= (216,216) \\ &= 216 \end{align*}

Since \frac{1944}{216} = 9 and \frac{864}{216} = 4, this problem becomes equivalent to the one of finding the probability of randomly chosen positive integer multiples of 4 is divisible by 9. Since 4 and 9 are coprime, we only i\times 4\times 9 can be considered. Hence, that probability is \frac{1}{9}. \blacksquare

Posted By Lee

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