1. **State the problem:** We need to divide the binary number $111110000001_2$ by $111111_2$ using long division in base 2. 2. **Recall the division process:** Binary long division works similarly to decimal long division. We compare the divisor to the current portion of the dividend, subtract if possible, and bring down the next bit. 3. **Convert divisor and dividend to decimal for clarity:** - Dividend: $111110000001_2 = 3969_{10}$ - Divisor: $111111_2 = 63_{10}$ 4. **Perform decimal division to find quotient and remainder:** $$3969 \div 63 = 63 \text{ remainder } 0$$ 5. **Convert quotient back to binary:** $$63_{10} = 111111_2$$ 6. **Therefore, the quotient is $111111_2$ and the remainder is $0_2$.** 7. **Verification:** $$111111_2 \times 111111_2 = 111110000001_2$$ This confirms the division is exact. **Final answer:** $$\boxed{111111_2}$$