Find the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case.

### Concept & Formula When asked to find the greatest number that divides $x, y,$ and $z$ leaving the same remainder in each case, the required number is given by taking the HCF of their absolute differences: $$ \text{Required Number} = \text{HCF of } (|x - y|, |y - z|, |z - x|) $$ ### Step-by-Step Solution * **Given:** Numbers are 43, 91, and 183. * Step 1: Find the absolute differences between the given numbers pairwise. $$91 - 43 = 48$$ $$183 - 91 = 92$$ $$183 - 43 = 140$$ * Step 2: Calculate the HCF of 48, 92, and 140. * Let's break them down: $$48 = 4 \times 12$$ $$92 = 4 \times 23$$ $$140 = 4 \times 35$$ * The numbers 12, 23, and 35 have no common prime factors. * Therefore, the highest common factor among these differences is 4. ### Exam Strategy & Shortcut **Option Elimination** is vastly faster here! Instead of doing the difference and HCF calculations, just plug the options into the original numbers to check for identical remainders. Dividing by 4 gives a remainder of 3 in all cases ($43 = 40+3$, $91 = 88+3$, $183 = 180+3$). You can verify this in under 10 seconds. ### Common Pitfall A standard trap is trying to find the HCF of 43, 91, and 183 directly. This won't work because they don't share a common factor, and it completely ignores the "same remainder" constraint. Always remember to take the differences first for this specific question type. ### Final Answer **Therefore, the correct answer is 4.**