Finding the Least Perfect Square Divisible by 21, 36 and 66

Given: 21, 36, and 66 are three numbers

To find: The least perfect square that is divisible by all three numbers

Solution:

Step 1: Find the prime factorization of all three numbers

- 21 = 3 × 7
- 36 = 2 × 2 × 3 × 3
- 66 = 2 × 3 × 11

Step 2: Identify the common factors and their highest power in all three numbers

- 2 × 3 = 6
- 3
- There is no common factor of 7 and 11.

Step 3: Multiply the common factors with their highest power to get the LCM (Least Common Multiple)

- LCM of 21, 36 and 66 = 2 × 3 × 3 × 7 × 11 = 1386

Step 4: Find the square of the LCM

- (1386)² = 1,920,996

Step 5: Check if the square obtained in step 4 is divisible by all three numbers

- 1,920,996 is not divisible by 21
- 1,920,996 is not divisible by 36
- 1,920,996 is divisible by 66

Step 6: Increment the LCM obtained in step 3 by the smallest prime factor (3) until we get a perfect square that is divisible by all three numbers

- LCM + 3 = 1389
- (1389)² = 1,934,121
- 1,934,121 is divisible by 21, 36, and 66

Step 7: Verify that the obtained square is the least possible

- To verify that 1,934,121 is the least perfect square that is divisible by 21, 36, and 66, we can check all the perfect squares that are greater than 1,934,121 and less than or equal to (1389)² = 1,934,121.
- The next perfect square after 1,934,121 is (1389 + 1)² = 1,937,764, which is not divisible by 21, 36, and 66.
- Therefore, the least perfect square that is divisible by 21, 36, and 66 is (1389)² = 1,934,121.

Answer: (A) 213444