Here is the full solution, given that negative temperatures are allowed.
Rearrange the conversion formula, getting 160 = 5F - 9C
Part 1) F = x, c = nx
160 = 5x - 9nx
x = 160/(5 - 9n)
The divisors of 160 are
1, 2, 4, 8, 16 and 32,
and 5, 10, 20, 40, 80, 160
and -1, -2, -4, -8, -16, -32,
and -5, -10, -20, -40, -80, -160
We are looking for divisors which equal 5 mod 9,
and the only ones are 32, 5, -4, and -40
5 - 9n = 32 gives
n = -3, x = 5 = F, C = -15 5 - 9n = 5 gives n = 0, x = 32 = F, C = 0
5 - 9n = -4 gives
n = 1, x = -40 = F, C = -40 5 - 9n = -40 gives
n = 5, x = -4 = F, C = -20
Part 1) F = nx, c = x
160 = 5nx - 9x
x = 160/(5n - 9)
5n - 9 is not a multiple of 5.
The only divisors of 160 that are not a multiple of 5 are
1, 2, 4, 8, 16 and 32,
and -1, -2, -4, -8, -16, -32
5n - 9 = 1 (mod 5),
and the only divisors that equal 1 mod 5
are 1 and 16 and -4
5n - 9 = 1 gives n = 2, x = 160 = C, F = 320
5n - 9 = 16 gives n = 5, x = 10 = C, F = 50
5n - 9 = -4 gives n = 1, x = -40 = C, F = -40
Baby, it's cold and warm outside (spoiler)
