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What is the probability that a randomly selected divisor of 720 is a multiple of 4? Express your answer as a common fraction.

 Apr 3, 2021
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Listing out the divisors of 720, we have:

 

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 36, 40, 45, 48, 60, 72, 80, 90, 120, 144, 180, 240, 360, and 720.

 

The divisors of 720 that are a multiple of 4 are:

 

4, 8, 12, 16, 20, 24, 36, 40, 48, 60, 72, 80, 120, 144, 180, 240, 360, 720

 

There are 30 divisors of 720, and 18 of them are a multiple of 4, so the probability of picking a divisor of 720 that is a multiple of 4 is $\frac{18}{30}=\boxed{\frac{3}{5}}$

 

However, this method is very time consuming, and any algebraic solution would be much appreciated.

 Apr 3, 2021

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