Determine all possible pairs (x,y) of
nonzero integers that satisfy this equation:
(x/y)oC = (x-y)oF Note:
oF = (9/5)*
oC+32, where
oF represents degrees Fahrenheit and,
oC represents degrees Celsius.
The intermediate goal will to create an equation where we have a product of linear terms set equal to an integer constant. Then we will break the product into two linear equations for each integer factorization of the integer constant.
Start with the obvious and substitute into the F-to-C formula:
x-y = (9/5)*(x/y)+32
Now move everything to one side and multiply through by 25y to get
(5y) * (5x-5y) - 45x - 800y = 0
Now we want to absorb the linear terms into the product, so if b and c are those values then
(5y + b) * (5x-5y + c) = bc
Which means -45x - 800y = 5bx-5by+5cy, or after equating coefficients and solving b=-9 and c=-169
So now (5y - 9) * (5x-5y - 151) = 1521 = 3^2*13^2 = f*g.
For each of the 18 signed factorizations of 1359 we want to solve
5y - 9 = f
5x - 5y - 151 = g
From the first linear equation we can conclude that f=1 mod 5 to have integer solutions. That reduces the number of cases to five: (f,g) = (1, 1521), (1521, 1), (-9, -169), (-169, -9), (-39, -39).
From those five cases we have (x,y) = (340, 2), (340, 306), (0,0), (-32, 0), (20, -6). After discarding zeros we are left with three answers: (x,y) = (340, 2), (340, 306), or (20, -6).
Analytic Solution
