examday

/TMUA

For a positive integer $n$, let $d(n)$ denote the number of positive divisors of $n$. For example $d(6)=4$, since $6$ has divisors $1$, $2$, $3$ and $6$.

Call $n$ compact if and only if $d(n)=4$.

Which of the following statements is/are true?

• I Every compact number is even.
• II Every compact number is the product of two distinct primes.
• III If $n$ is compact then $2n$ is compact.