Find the greatest number that will divide 964, 1238 and 1400 leaving remainders 41, 31 and 51 respectively.

### Concept & Formula When asked to find the greatest number that divides given numbers $x$, $y$, and $z$ leaving specific remainders $a$, $b$, and $c$ respectively, the required number is the Highest Common Factor (HCF) of the numbers obtained by subtracting the remainders from the original numbers. Required Number = $\text{HCF of } (x - a), (y - b), \text{ and } (z - c)$ ### Step-by-Step Solution **Given:** * Numbers: $964$, $1238$, and $1400$ * Respective Remainders: $41$, $31$, and $51$ **Calculation:** * Step 1: Subtract the respective remainders from the given numbers. * $964 - 41 = 923$ * $1238 - 31 = 1207$ * $1400 - 51 = 1349$ * Step 2: Find the HCF of $923$, $1207$, and $1349$. * Step 3: Use the difference method to find the HCF easily. Find the differences between adjacent numbers. * $1207 - 923 = 284$ * $1349 - 1207 = 142$ * Step 4: The HCF must be a factor of the smallest difference ($142$). * The factors of $142$ are $1, 2, 71, 142$. * Step 5: Check these factors against our numbers. Since $923, 1207, 1349$ are all odd, the HCF cannot be even ($2$ or $142$). Let's check $71$. * $923 \div 71 = 13$ * $1207 \div 71 = 17$ * $1349 \div 71 = 19$ * Since $71$ perfectly divides all three, the HCF is $71$. ### Exam Strategy & Shortcut The difference method for finding HCF is your best friend here. Instead of tedious long division, simply find the differences between the numbers: $(1207 - 923) = 284$ and $(1349 - 1207) = 142$. The HCF must be a factor of these differences. Noting the numbers are odd immediately eliminates $142$ and $2$, leaving $71$ as the obvious prime candidate. You can also quickly verify $71$ by comparing it directly against the provided options! ### Common Pitfall Students often attempt to find the HCF of the original numbers ($964, 1238, 1400$) and then adjust for the remainders afterward. This mathematical logic is flawed and will always yield an incorrect option. Always subtract the remainders *before* calculating the HCF. ### Final Answer **Therefore, the correct answer is 71.**