If d is the HCF of 468 and 222, find the value of integers x and y which satisfy the equation 468 = d × x and 222 = d × y.

To find integers x and y such that 468 = d × x and 222 = d × y, we first need to determine the highest common factor (HCF) of the numbers 468 and 222.

We can calculate the HCF using the prime factorization method or the Euclidean algorithm. For simplicity, let’s use the Euclidean algorithm:

1. Divide 468 by 222, which gives a quotient of 2 and a remainder of 24. (468 = 222 × 2 + 24)
2. Next, divide 222 by 24, which gives a quotient of 9 and a remainder of 6. (222 = 24 × 9 + 6)
3. Now, divide 24 by 6, which gives a quotient of 4 and a remainder of 0. (24 = 6 × 4 + 0)

Since the last non-zero remainder is 6, the HCF (d) of 468 and 222 is 6.

Now that we have d = 6, we can substitute this value back into the equations 468 = d × x and 222 = d × y to find x and y:

1. For the first equation: 468 = 6 × x. Dividing both sides by 6 gives us x = 468 / 6 = 78.
2. For the second equation: 222 = 6 × y. Dividing both sides by 6 gives us y = 222 / 6 = 37.

Thus, the integers x and y that satisfy the equations are:

• x = 78
• y = 37