What is the total number of valid codes for lockers that are either all even or all odd-prime based on the conditions given?

To determine the total number of valid codes for lockers that are either all even or all odd-prime, it's essential to analyze the requirements laid out.

Starting with even numbers, the only even prime number is 2. Therefore, an all even-prime code can only consist of the digit 2. If the code's length is not specified, let's assume we can form a code of a specific length, represented as ( n ). Consequently, the total possibilities for an all-even code would be limited to a single combination: '222...2' (where '2' is repeated for the length of the code). If the length permits any repetitions, however, we cannot generate a varying combination from only one digit.

Now focusing on odd primes, the odd prime numbers up to 10 are 3, 5, and 7. Each of these can be combined to form code combinations of a specific length, assuming that we are generating codes using these digits only. For any position in the code of length ( n ), we can choose from these three odd primes (3, 5, or 7). Hence, for a code of length ( n ), the total combinations from the odd primes can be calculated as (