ind the least square number which is divisible by 6, 8 and 15Options300360037003900

Step 1: Break Down the Problem

To find the least square number that is divisible by 6, 8, and 15, we need to:

1. Determine the least common multiple (LCM) of the numbers 6, 8, and 15.
2. Find the smallest perfect square that is divisible by this LCM.

Step 2: Relevant Concepts

The LCM of several numbers can be found using their prime factorization. The prime factorization of each number is:

• $6 = 2^1 \cdot 3^1$
• $8 = 2^3$
• $15 = 3^1 \cdot 5^1$

The LCM is found by taking the highest power of each prime that appears in the factorizations.

Step 3: Analysis and Detail

1. Calculating LCM: $\text{LCM}(6, 8, 15) = 2^{\max(1, 3, 0)} \cdot 3^{\max(1, 0, 1)} \cdot 5^{\max(0, 0, 1)} = 2^3 \cdot 3^1 \cdot 5^1$ $= 8 \cdot 3 \cdot 5 = 120$

2. Finding the Smallest Perfect Square: To find the least square number divisible by $120$, we need to express $120$'s prime factorization: $120 = 2^3 \cdot 3^1 \cdot 5^1$ For a number to be a perfect square, all prime factors must have even exponents. We have:

   • $2^3$: needs one more $2$ to make it $2^4$,
   • $3^1$: needs one more $3$ to make it $3^2$,
   • $5^1$: needs one more $5$ to make it $5^2$.

   Therefore, to form a perfect square that is divisible by $120$, we must multiply $120$ by $2^1 \cdot 3^1 \cdot 5^1$: $120 \cdot (2^1 \cdot 3^1 \cdot 5^1) = 120 \cdot 30 = 3600$

Step 4: Verify and Summarize

• The LCM of $6$, $8$, and $15$ is $120$.
• The least square number that is divisible by $120$ is $3600$.

Final Answer

The least square number which is divisible by 6, 8, and 15 is 3600.