1. **Problem statement:** Given a message 1110001 and a generator polynomial 111, find the CRC remainder. 2. **Formula and rules:** CRC remainder is found by dividing the message appended with zeros (length of generator minus 1) by the generator using binary division (XOR operation). 3. **Step 1:** Append two zeros to the message (since generator length is 3, append 3-1=2 zeros): $$1110001\,00 = 111000100$$ 4. **Step 2:** Perform binary division (XOR) of 111000100 by 111: - Divide first 3 bits: 111 XOR 111 = 000 - Bring down next bit: 0, new bits: 000 - Since leading bit is 0, bring down next bit: 0, new bits: 000 - Still leading bit 0, bring down next bit: 1, new bits: 001 - Leading bit 0, bring down next bit: 0, new bits: 010 - Leading bit 0, bring down next bit: 0, new bits: 100 - Leading bit 1, XOR with 111: 100 XOR 111 = 011 - Bring down last bit: 0, new bits: 110 - Leading bit 1, XOR with 111: 110 XOR 111 = 001 5. **Step 3:** The remainder is the last two bits (length of generator minus 1): 01 6. **Answer:** The CRC remainder is **01**.