Find the smallest number of five digits exactly divisible by $16$, $24$, $36$ and $54$.

### Concept & Strategy To find the smallest $N$-digit number divisible by a set of numbers, you divide the base smallest $N$-digit number (e.g., $10000$) by the L.C.M. of the divisors. Then, add the difference between the L.C.M. and the remainder to the base number to push it up to the next perfect multiple. $$ \text{Required Number} = \text{Smallest } N\text{-digit} + (\text{L.C.M.} - \text{Remainder}) $$ ### Step-by-Step Solution * **Given:** Divisors are $16$, $24$, $36$, and $54$. Target is the smallest 5-digit number. * The smallest 5-digit number is $10000$. * Find the L.C.M. of $16$, $24$, $36$, and $54$: * $16 = 2^4$ * $24 = 2^3 \times 3$ * $36 = 2^2 \times 3^2$ * $54 = 2 \times 3^3$ * $\text{L.C.M.} = 2^4 \times 3^3 = 16 \times 27 = 432$ * Divide $10000$ by the L.C.M. ($432$): * $10000 \div 432$ yields a quotient of $23$ and a remainder of $64$. * To make $10000$ perfectly divisible without dropping down to a 4-digit number, add the necessary difference: * $\text{Amount to add} = 432 - 64 = 368$ * $\text{Required Number} = 10000 + 368 = 10368$ ### Exam Strategy & Shortcut Use **Divisibility Rules**. The correct answer must be divisible by $54$, which implies it must be completely divisible by $9$. Check the digit sum of the options: * $10000 \rightarrow$ sum is $1$ (fail) * $10264 \rightarrow$ sum is $13$ (fail) * $10368 \rightarrow$ sum is $18$ (**pass**) * $10432 \rightarrow$ sum is $10$ (fail) You can find the answer in $5$ seconds using this check. ### Common Pitfall The most frequent mistake is simply subtracting the remainder ($64$) from the base number ($10000$). Doing so yields $9936$, which correctly divides by the L.C.M. but violates the core constraint of being a 5-digit number. ### Final Answer **Therefore, the correct answer is 10368.**