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 $n?$

cooIcooIcooI17 Jul 7, 2024

#1**+1 **

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! :)

NotThatSmart Jul 7, 2024