Number of Integral Solutions of the Equation (407)x - (ddd)y = 2589

To find the total number of integral solutions of the equation (407)x - (ddd)y = 2589, where 'ddd' is a three-digit number, we need to consider the possible values of 'ddd' and the conditions for integral solutions.

Values of 'ddd'

Since 'ddd' represents a three-digit number, it can take any value from 100 to 999.

Conditions for Integral Solutions

For the equation to have integral solutions, the values of x and y should be integers that satisfy the equation.

We can rearrange the equation to solve for y:

(407)x - (ddd)y = 2589
(ddd)y = (407)x - 2589
y = ((407)x - 2589)/(ddd)

For y to be an integer, the numerator of the right-hand side must be divisible by 'ddd'.

Case 1: ddd is divisible by 3

If 'ddd' is divisible by 3, then any value of x will result in an integral value of y. This is because (407)x - 2589 will always be divisible by 3, and dividing by 'ddd' will yield an integer value of y.

In this case, there are infinite integral solutions for any value of 'ddd' that is divisible by 3.

Case 2: ddd is not divisible by 3

If 'ddd' is not divisible by 3, then for there to be integral solutions, (407)x - 2589 must be divisible by 'ddd'. This restricts the possible values of x.

Let's consider a few examples to illustrate this:

1. If 'ddd' = 101, then (407)x - 2589 must be divisible by 101. The values of x that satisfy this condition will result in integral solutions for y.

2. If 'ddd' = 103, then (407)x - 2589 must be divisible by 103. Similarly, the values of x that satisfy this condition will give integral solutions for y.

We need to check the divisibility of (407)x - 2589 by 'ddd' for each possible value of 'ddd' from 100 to 999.

Conclusion

In summary, the total number of integral solutions of the equation (407)x - (ddd)y = 2589 depends on the divisibility of (407)x - 2589 by 'ddd'. If 'ddd' is divisible by 3, there are infinite integral solutions. If 'ddd' is not divisible by 3, the number of integral solutions depends on the divisibility of (407)x - 2589 by 'ddd' for each possible value of 'ddd' from 100 to 999.

Infinite