To determine the greatest number of name tag stations Pam can set up, we need to find the greatest common divisor (GCD) of the number of name tags and the number of pens she has, which are 48 and 45 respectively.
We can start by finding the prime factorization of each number.
Prime factorization of 48: \[ 48 = 2^4 \times 3^1 \]
Prime factorization of 45: \[ 45 = 3^2 \times 5^1 \]
Now, we identify the common prime factors. The only common prime factor is \(3\).
To find the GCD, we take the lowest power of the common prime factors:
- For the prime factor \(3\), the lowest power in 48 is \(3^1\) and in 45 is \(3^2\). Hence, we take \(3^1\).
Therefore, the GCD of 48 and 45 is: \[ GCD = 3^1 = 3 \]
Thus, the greatest number of name tag stations that Pam can set up without having any name tags or pens left over is: \[ \boxed{3} \]