1. **State the problem:** We need to find the CRC remainder when the generator polynomial is 1001 and the message is 1110. 2. **Explain the formula and rules:** CRC (Cyclic Redundancy Check) remainder is found by dividing the message polynomial (augmented with zeros equal to the degree of the generator) by the generator polynomial using binary division (XOR operations). 3. **Set up the problem:** - Generator (divisor) = 1001 (degree 3) - Message = 1110 - Append 3 zeros to message: 1110 000 = 1110000 4. **Perform binary division (XOR):** - Divide 1110000 by 1001 Step-by-step division: - 1110 XOR 1001 = 0111 (bring down next bit 0) -> 1110 - 1110 XOR 1001 = 0111 (bring down next bit 0) -> 1110 - 1110 XOR 1001 = 0111 (bring down next bit 0) -> 1110 Actually, let's do it carefully: Initial dividend: 1110000 - Take first 4 bits: 1110 - 1110 XOR 1001 = 0111 - Bring down next bit: 0 -> 1110 - 1110 XOR 1001 = 0111 - Bring down next bit: 0 -> 1110 - 1110 XOR 1001 = 0111 - Bring down next bit: 0 -> 1110 This seems repetitive, so let's do the division properly: Dividend: 1110000 Step 1: 1110 XOR 1001 = 0111 Bring down next bit: 0 -> 1110 Step 2: 1110 XOR 1001 = 0111 Bring down next bit: 0 -> 1110 Step 3: 1110 XOR 1001 = 0111 Bring down next bit: 0 -> 1110 Since the message length is 7 bits and generator degree is 3, the remainder is the last 3 bits after division. But this is confusing, so let's do the division in a clearer way: Dividend: 1110000 - Align divisor 1001 under first 4 bits: 1110 1001 ---- 0111 Bring down next bit: 0 -> 1110 - Align divisor under 4 bits: 1110 1001 ---- 0111 Bring down next bit: 0 -> 1110 - Align divisor under 4 bits: 1110 1001 ---- 0111 Bring down next bit: 0 -> 1110 - Align divisor under 4 bits: 1110 1001 ---- 0111 No more bits to bring down. The remainder is 111 (last 3 bits). 5. **Final answer:** The CRC remainder is **111**. This means the remainder after dividing the message 1110 (with appended zeros) by the generator 1001 is 111.