1. **State the problem:** Divide the base-4 number $(321230)_4$ by $(123)_4$ and show the whole process. 2. **Convert both numbers from base 4 to base 10:** - For $(321230)_4$, calculate each digit times $4^n$ where $n$ is the position from right starting at 0: $$3 \times 4^5 + 2 \times 4^4 + 1 \times 4^3 + 2 \times 4^2 + 3 \times 4^1 + 0 \times 4^0$$ Calculate powers: $$3 \times 1024 + 2 \times 256 + 1 \times 64 + 2 \times 16 + 3 \times 4 + 0 \times 1$$ Calculate each term: $$3072 + 512 + 64 + 32 + 12 + 0 = 3692$$ So, $(321230)_4 = 3692_{10}$. - For $(123)_4$: $$1 \times 4^2 + 2 \times 4^1 + 3 \times 4^0 = 1 \times 16 + 2 \times 4 + 3 \times 1 = 16 + 8 + 3 = 27$$ So, $(123)_4 = 27_{10}$. 3. **Perform the division in base 10:** $$3692 \div 27 = 136 \text{ remainder } 20$$ 4. **Convert the quotient and remainder back to base 4:** - Quotient 136 to base 4: Divide 136 by 4 repeatedly: $$136 \div 4 = 34 \text{ remainder } 0$$ $$34 \div 4 = 8 \text{ remainder } 2$$ $$8 \div 4 = 2 \text{ remainder } 0$$ $$2 \div 4 = 0 \text{ remainder } 2$$ Reading remainders from last to first: $(2200)_4$ - Remainder 20 to base 4: $$20 \div 4 = 5 \text{ remainder } 0$$ $$5 \div 4 = 1 \text{ remainder } 1$$ $$1 \div 4 = 0 \text{ remainder } 1$$ Reading remainders: $(110)_4$ 5. **Final answer:** $$\frac{(321230)_4}{(123)_4} = (2200)_4 \text{ remainder } (110)_4$$