degrees = adc10 / K

so for 359 = 1023 / K

K = 1023/359

K= 2.8495821727019498607242339832869

Now you need to make the integer divisor as big as possible to reduce rounding errors, eg multiply by N, but you also need to multiply adc10 by N so the result is the same ...

degree = ( adc10 * N ) / (K * N )

Having N as big as possible reduces error but N cannot be bigger than 64 as (1023*65) is an overflow of 16-bit ( greater than 65535 ) so let's use N = 64, K*N = 182, so ...

degree = adc10 * 64 / 182

That's at least a good first pass. Someone better at maths may be able to work out a more accurate away of approaching it !