Question

How many positive integer pairs (x,y) satisfy 4x+12y=640?

Answer 1

There is only one positive integer pair (x, y) that satisfies the equation 4x + 12y = 640.

Step-by-step explanation:

To solve the equation 4x + 12y = 640, we need to find the number of positive integer pairs (x, y) that satisfy the equation.

We can start by dividing both sides of the equation by 4 to simplify it: x + 3y = 160.

Next, we can rewrite the equation in terms of y: y = (160 - x) / 3.

To find the positive integer pairs, we can substitute different values of x into the equation and check if the corresponding value of y is a positive integer.

Let's try some values of x:

1. When x = 1, y = (160 - 1) / 3 = 53/3, which is not an integer.
2. When x = 2, y = (160 - 2) / 3 = 158/3, which is not an integer.
3. When x = 3, y = (160 - 3) / 3 = 157/3, which is not an integer.
4. ...
5. When x = 157, y = (160 - 157) / 3 = 1, which is a positive integer.
6. When x = 158, y = (160 - 158) / 3 = 0, which is not a positive integer.
7. When x = 159, y = (160 - 159) / 3 = 1/3, which is not an integer.
8. When x = 160, y = (160 - 160) / 3 = 0, which is not a positive integer.

Therefore, there is only one positive integer pair (x, y) that satisfies the equation 4x + 12y = 640.