Find the largest number which when subtracted from $10000$, the remainder is divisible by $32$, $36$, $48$ and $54$.

### Concept & Logic Let the required number be $X$. The problem states that when $X$ is subtracted from $10000$, the resulting difference is completely divisible by a given set of numbers. This means the difference must be the L.C.M. of those divisors (to ensure $X$ is as large as possible, we subtract the smallest possible multiple, which is the L.C.M.). $$ 10000 - X = \text{L.C.M.}(a, b, c, d) $$ $$ X = 10000 - \text{L.C.M.}(a, b, c, d) $$ ### Step-by-Step Solution * **Given:** Base number $10000$. Divisors are $32$, $36$, $48$, and $54$. * First, find the L.C.M. of the divisors using prime factorization: * $32 = 2^5$ * $36 = 2^2 \times 3^2$ * $48 = 2^4 \times 3$ * $54 = 2 \times 3^3$ * Extract the highest powers of the prime factors: * For $2$: $2^5 = 32$ * For $3$: $3^3 = 27$ * Multiply to find the L.C.M.: * $\text{L.C.M.} = 32 \times 27 = 864$ * Now apply the core logic equation. We know $10000 - X = 864$. * $X = 10000 - 864 = 9136$ ### Exam Strategy & Shortcut You can work directly from the options using divisibility rules. The question asks for an option $X$ such that $(10000 - X)$ is a multiple of $54$ (and thus a multiple of $9$). * $10000 - 9136 = 864$ (Digit sum is $18 \rightarrow$ Divisible by $9$) * $10000 - 864 = 9136$ (Digit sum is $19 \rightarrow$ Fail) Option B successfully creates a difference that satisfies the divisibility condition. ### Common Pitfall A very common trap is doing all the hard work to calculate the L.C.M. ($864$) and then immediately choosing $864$ as the final answer. You must re-read the prompt carefully: it asks for the number *subtracted from* $10000$ to get the L.C.M., not the L.C.M. itself. ### Final Answer **Therefore, the correct answer is 9136.**