Determine all possible values of a positive integer x satisfying:

x degrees Fahrenheit = sod(x) degrees Celsius, where sod(x) denotes the sum of the digits in the base ten expansion of x.

Prove that there are no others.

Note: To convert from degrees Fahrenheit to Celsius, subtract 32 then multiply by 5/9. To convert the other way, simply do the opposite (multiply by 9/5 and add 32).