To find out how often the librarian restocks both books and magazines on the same day, we need to calculate the least common multiple (LCM) of the two restocking periods: 12 days and 18 days.
First, we can find the prime factorization of each number:
- \(12 = 2^2 \times 3^1\)
- \(18 = 2^1 \times 3^2\)
Next, we take the highest power of each prime factor from both factorizations:
- For the prime number 2, the highest power is \(2^2\).
- For the prime number 3, the highest power is \(3^2\).
Now we calculate the LCM:
\[ \text{LCM} = 2^2 \times 3^2 = 4 \times 9 = 36 \]
Thus, the librarian will restock both books and magazines on the same day every 36 days.