At the brand new 'Prime Portal' ride in the amusement park, a unique entry rule applies: only visitors whose ticket numbers *cannot* be expressed as the product of *exactly two different single-digit prime numbers* are permitted to enter. Four friends, Alex, Ben, Chloe, and Dana, each have a ticket. Which friend's ticket number would allow them to enter the Prime Portal?
Alex's ticket: 14
Ben's ticket: 15
Chloe's ticket: 12
Dana's ticket: 21
Correct: Chloe
First, let's identify the single-digit prime numbers: 2, 3, 5, and 7.
Next, according to the rule, we need to find all possible products of *exactly two different* single-digit prime numbers. These are the ticket numbers that are *forbidden* from entering the ride:
* 2 × 3 = 6
* 2 × 5 = 10
* 2 × 7 = 14
* 3 × 5 = 15
* 3 × 7 = 21
* 5 × 7 = 35
So, ticket numbers 6, 10, 14, 15, 21, and 35 are NOT allowed into the 'Prime Portal'.
Now, let's check each friend's ticket number against this list:
* **Alex's ticket: 14**. This is the product of 2 × 7. So, Alex is **NOT allowed**.
* **Ben's ticket: 15**. This is the product of 3 × 5. So, Ben is **NOT allowed**.
* **Chloe's ticket: 12**. Can 12 be expressed as the product of exactly two different single-digit prime numbers? No. Its prime factorization is 2 × 2 × 3 (which involves three primes, and two of them are the same). It cannot be formed by multiplying just two *different* single-digit primes. So, Chloe **IS allowed**.
* **Dana's ticket: 21**. This is the product of 3 × 7. So, Dana is **NOT allowed**.
Therefore, Chloe is the only friend whose ticket number (12) allows her to enter the Prime Portal.