Final answer:
To find the pairs of numbers with an HCF of 3 and an LCM of 45, we calculate the product of these values and then determine the pairs of factors of this product that also have an HCF of 3. The only pairs that meet these criteria are (3,45) and (9,15).
Step-by-step explanation:
Finding All Possible Pairs of Numbers with Given HCF and LCM
The question asks us to find all possible pairs of numbers where the highest common factor (HCF) is 3, and the lowest common multiple (LCM) is 45. The product of two numbers is equal to the product of their HCF and LCM. So, the product of our two numbers is 3 × 45 = 135.
To find the pairs of numbers, we need to find all pairs of factors of 135 that also have an HCF of 3. The factors of 135 that can be paired are (1,135), (3,45), (5,27), and (9,15). However, not all these pairs will have an HCF of 3. Only the pairs (3,45) and (9,15) meet both criteria, as the other pairs either don't have an HCF of 3 or involve the repetition of the same two numbers in different order.