+821 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 $n?$
+1946 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! :)
