1. **State the problem:** Find the greatest number that divides 43, 91, and 183 leaving the same remainder in each case. 2. **Key idea:** If a number $d$ divides these numbers leaving the same remainder, then $d$ divides the differences of these numbers. 3. Calculate the differences: $$91 - 43 = 48$$ $$183 - 91 = 92$$ $$183 - 43 = 140$$ 4. Find the greatest common divisor (GCD) of 48, 92, and 140. 5. Find $\gcd(48, 92)$: $$48 = 2 \times 24$$ $$92 = 2 \times 46$$ Common factor is 2. 6. Use Euclidean algorithm: $$\gcd(48, 92) = \gcd(48, 92 - 48) = \gcd(48, 44)$$ $$\gcd(48, 44) = \gcd(48 - 44, 44) = \gcd(4, 44) = 4$$ 7. Now find $\gcd(4, 140)$: $$\gcd(4, 140) = 4$$ 8. Therefore, the greatest number that divides 43, 91, and 183 leaving the same remainder is $\boxed{4}$.