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A positive integer is called nice if it is a multiple of $8.$
A certain nice positive integer $n$ has exactly $9$ positive divisors. How many prime numbers are divisors of $n?$

 Jul 7, 2024
 #1
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First, let's note that if a number has an odd number of factors, then it must be a perfect square. 

In the problem, it also must be divisible by 8. 

 

Thus first few that satsify this number is \(16,64,144\)

 

However, none of them work, as 16 and 64 have too few factors, and 144 has too many. 

The next number is 256. 

 

Note that 256 has exactly 9 factors of \(1, 2, 4, 8, 16, 32, 64, 128,256\)

 

Of these 9 factors, \(2\) is the only prime divisor!

 

So out of the 9, only 1 is prime

 

So 1 is our answer.

 

Thanks! :)

 Jul 7, 2024

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