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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. What is the smallest possible value of $n?$

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

It also must be a multiple of 8.

 

So we need a perfect square divisble by 8. 

First, we have

\(4*4=16\)

 

However, 16 only has 5 factors, with \(1,2,4,8,16\)

 

Next, we have \(8*8=64\). However, 64 only has 7 factors, with \(1, 2, 4, 8, 16, 32, 64\)

144 doesn't work, as it has way too much. 

 

However, we note that \(16*16=256\)

 

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

 

Thus, the smallest possible number is 256. 

So 256 is our answer. 

 

Thanks! :)

 Jul 7, 2024

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