[Number Theory]
Problem: The positive integers 34 and 80 have exactly two positive common divisors, namely 1 and 2. How many positive integers $n$ with $1 \leq n \leq 30$ have the property that $n$ and 80 have exactly two positive common divisors?

Solution: Since $80=2^{4} \cdot 5$, its positive divisors are $1,2,4,5,8,10,16,20,40,80$.

For an integer $n$ to share exactly two positive common divisors with 80, these divisors must be either 1 and 2 or 1 and 5 . ( 1 is a common divisor of any two integers. The second common divisor must be a prime number since any composite divisor will cause there to be at least one more common divisor which is prime.)

Since $1 \leq n \leq 30$ and $n$ is a multiple of 2 or of 5 , then the possible values of $n$ come from the list

$$
2,4,5,6,8,10,12,14,15,16,18,20,22,24,25,26,28,30
$$

We remove the multiples of 4 from this list (since they would share at least the divisors $1,2,4$ with 80 ) and the multiples of 10 from this list (since they would share at least the divisors $1,2,5,10$ with 80 ).

This leaves the list

$$
2,5,6,14,15,18,22,25,26
$$

The common divisors of any number from this list and 80 are either 1 and 2 or 1 and 5 . There are 9 such integers.

Final answer: 9