1. **Stating the problem:** We want to find the highest possible number of books you have, which is less than 75, and this number must be a prime number (divisible only by 1 and itself). 2. **Formula and rules:** A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. 3. **Approach:** We need to find the largest prime number less than 75. 4. **Check prime numbers less than 75:** The prime numbers less than 75 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73. 5. **Identify the highest prime less than 75:** The highest prime number less than 75 is 73. **Final answer:** The highest possible number of books you have is **$73$**.