1. **Problem Statement:** We have 30 dictionaries with a total of 61,327 pages. We need to prove that at least one dictionary has at least 2,045 pages. 2. **Formula and Principle Used:** This is a classic application of the Pigeonhole Principle, which states that if $n$ items are put into $k$ containers, then at least one container must contain at least $\lceil \frac{n}{k} \rceil$ items. 3. **Applying the Principle:** Here, the "items" are pages and the "containers" are dictionaries. 4. **Calculate the average number of pages per dictionary:** $$\text{Average pages} = \frac{61,327}{30} = 2044.2333...$$ 5. **Interpretation:** Since the average number of pages per dictionary is approximately 2044.23, if every dictionary had fewer than 2045 pages, the total pages would be less than: $$30 \times 2044 = 61,320$$ 6. **Contradiction:** But the total pages are 61,327, which is greater than 61,320. 7. **Conclusion:** Therefore, at least one dictionary must have at least 2,045 pages to reach the total of 61,327 pages. This completes the proof using the Pigeonhole Principle.