Find the greatest number that divides 43, 91, and 183 leaving the same remainder in each case.

Given data

• Same remainder when dividing 43, 91, 183 by the required number.

Concept / Approach

• Greatest such number equals GCD of pairwise differences.

Step-by-step

Differences: 91 − 43 = 48; 183 − 91 = 92; 183 − 43 = 140GCD(48, 92) = 4GCD(4, 140) = 4

Verification

43 mod 4 = 3; 91 mod 4 = 3; 183 mod 4 = 3. Same remainder.

Common pitfalls

• Taking GCD of the numbers themselves instead of their pairwise differences.

Final Answer

Greatest required number = 4.