[thaumasiotes]

As regards your third line, I feel compelled to note that if we know the equivalence class of two integers (mod 37), we also know the equivalence class (mod 37) of their sum and product. ;)

[kazinator]

Indeed, it just the mod 37 congruence doesn't correspond to nice Boolean:

  odd(x + y) = odd(x) XOR odd(y)
  odd(x * y) = odd(x) AND odd(y)

There.