Step 1: Understanding the Concept:
To solve this, we first need to find the Least Common Multiple (L.C.M.) of the numbers 6, 9, and 15, then determine the next perfect square.
Step 2: Key Formula or Approach:
- Find the prime factorization of each number:
\[ 6 = 2 \times 3 \]
\[ 9 = 3^2 \]
\[ 15 = 3 \times 5 \]
- L.C.M. = \( 2 \times 3^2 \times 5 = 90 \)
Step 3: Detailed Explanation:
- The smallest number divisible by 6, 9, and 15 is 90.
- We need to find the smallest perfect square greater than 90.
- The squares near 90 are \( 9^2 = 81 \) and \( 10^2 = 100 \).
- The smallest perfect square greater than 90 is 100.
- The number to be added is \( 100 - 90 = 10 \).
*Self-correction:* Re-evaluating the options provided, if 10 is the calculated answer and it is present in option (A), we ensure the logic holds. Let's re-verify the perfect square: 100 is the next one. 100 - 90 = 10.
Step 4: Final Answer:
The smallest number to be added is 10.