Two numbers have a sum of 528 and their highest common factor (HCF) is 33. How many distinct pairs of numbers satisfy these conditions?

Explanation

Let the two numbers be 33a and 33b, where a and b are co-prime integers. Since their sum is 528, we have 33a + 33b = 528, which simplifies to a + b = 16. The pairs of co-prime integers (a, b) that add up to 16 are (1, 15), (3, 13), (5, 11), and (7, 9). Multiplying these by 33 gives the original number pairs. Therefore, there are 4 such pairs.