The least perfect square number divisible by $3$, $4$, $5$, $6$ and $8$ is

### Concept & Strategy To find a perfect square divisible by multiple numbers, first find the Least Common Multiple (LCM) of those numbers. Then, prime factorize the LCM and multiply it by any prime factors needed to make all exponents even. ### Step-by-Step Solution Find the LCM of $3$, $4$, $5$, $6$, and $8$. List their prime factorizations: $3 = 3^1$ $4 = 2^2$ $5 = 5^1$ $6 = 2^1 \times 3^1$ $8 = 2^3$ The LCM takes the highest power of each prime present: LCM $= 2^3 \times 3^1 \times 5^1$ LCM $= 8 \times 3 \times 5 = 120$. Now, analyze the prime factorization of the LCM ($2^3 \times 3^1 \times 5^1$) for odd powers. The power of $2$ is $3$ (odd). The power of $3$ is $1$ (odd). The power of $5$ is $1$ (odd). To make it a perfect square, we need to multiply by one more of each prime factor to make their powers even ($2^4, 3^2, 5^2$). Multiplier needed $= 2 \times 3 \times 5 = 30$. Multiply the LCM by this required multiplier: Least Perfect Square $= 120 \times 30 = 3600$. ### Exam Strategy & Shortcut Instead of calculating the LCM fully, look at the options. The number must be a perfect square. Options (b) $1200$ and (c) $2500$ are not divisible by $3$, so eliminate them. Option (a) $900$ is a perfect square and divisible by $3, 4, 5, 6$, but it is NOT divisible by $8$ ($900 / 8 = 112.5$). This leaves only $3600$ by process of elimination. ### Common Pitfall A common error is stopping once the LCM ($120$) is found and selecting the nearest multiple from the options without ensuring the final result is a perfect square. ### Final Answer Therefore, the correct answer is **3600**.