1. **Problem statement:** We are given that the sum of two numbers is 528 and their highest common factor (HCF) is 33. We need to find the two numbers. 2. **Formula and rules:** If the HCF of two numbers is 33, then the numbers can be expressed as $33x$ and $33y$, where $x$ and $y$ are co-prime (their HCF is 1). 3. **Using the sum:** Given the sum is 528, we write: $$33x + 33y = 528$$ 4. **Simplify the equation:** $$33(x + y) = 528$$ $$x + y = \frac{528}{33} = 16$$ 5. **Find pairs $(x, y)$:** Since $x$ and $y$ are co-prime and their sum is 16, possible pairs are (1, 15), (3, 13), (5, 11), (7, 9), etc. We check which pairs are co-prime. 6. **Check co-primality:** All these pairs are co-prime. 7. **Find the actual numbers:** Multiply each pair by 33: - For (1, 15): numbers are $33 \times 1 = 33$ and $33 \times 15 = 495$ - For (3, 13): numbers are $99$ and $429$ - For (5, 11): numbers are $165$ and $363$ - For (7, 9): numbers are $231$ and $297$ 8. **Verify sum:** All pairs sum to 528. 9. **Conclusion:** The possible pairs of numbers are $(33, 495)$, $(99, 429)$, $(165, 363)$, and $(231, 297)$. **Final answer:** The two numbers can be any of these pairs: $33$ and $495$, $99$ and $429$, $165$ and $363$, or $231$ and $297$.