Final answer:
The values of x and y are found by examining their common prime factors and the powers in their HCF and LCM. The integer x is equal to 6125 and y is 33,075.
Step-by-step explanation:
Finding the Values of x and y
To find the values of x and y based on the given Highest Common Factor (HCF) and Least Common Multiple (LCM), we need to recognize that the HCF is the product of the common prime factors and the LCM is the product of the maximum power of all prime factors involved.
Since the HCF of x and y is 1225 and we know that 1225 = 5² × 7², we can deduce that the common prime factors of x and y are 5² and 7². For the LCM, which is 42875, we can factorize it to find that 42875 = 3³ × 5³ × 7². Since the LCM contains the highest powers of prime factors from both x and y, we can further deduce that x must contain 5³ as its highest power of 5, and y must contain 3³ as its highest power of 3, because 1225 (the HCF) already accounts for the 7² present in both.
Thus, we arrive at x being (5² × 7²) × 5 = 5³ × 7² and y being (5² × 7²) × 3³ = 3³ × 5² × 7². Therefore, x = 5³ × 7² = 125 × 49 = 6125 and y = 3³ × 5² × 7² = 27 × 25 × 49 = 33,075.