Find the greatest number of $4$ digits which when divided by $4$, $5$, $6$, $7$ and $8$ leaves $1$, $2$, $3$, $4$ and $5$ as remainders.

### Concept & Strategy Common difference $K = \text{Divisor} - \text{Remainder} = 3$. Formula: $N = \text{Multiple of LCM} - 3$. ### Step-by-Step Solution * **Find LCM of $(4, 5, 6, 7, 8)$:** $\text{LCM} = 840$. * **Greatest 4-digit number:** $9999 \div 840 = 11$ remainder $759$. $9999 - 759 = 9240$. * **Apply constant difference:** $9240 - 3 = 9237$. ### Exam Strategy & Shortcut Check options: $(Option + 3)$ must be divisible by $840$. $9237 + 3 = 9240$. $9240 \div 840 = 11$. Correct. ### Common Pitfall Forgetting the final subtraction of $3$. Always subtract the common difference $K$ at the end. ### Final Answer **Therefore, the correct answer is 9237.**