One value, several representations
Binary, octal, decimal, and hexadecimal are different ways to write the same value. Addition and subtraction follow the same positional rules, but carrying happens at the base: at 2 in binary, 8 in octal, 10 in decimal, and 16 in hexadecimal.
Hexadecimal uses A through F for digit values 10 through 15. One hexadecimal digit corresponds to four binary bits, while one octal digit corresponds to three binary bits. That makes both bases convenient compact forms of binary data.
Division and remainders
Integer division produces a quotient and, when the division is not exact, a remainder. For example, 10₂ divided by 11₂ is 0 with remainder 10₂. A calculator should state that integer behavior instead of silently discarding the remainder.