The least perfect square, which is divisible by each of 21, 36 and 66 is:a)213444b)214344c)214434d)231444Correct answer is option 'A'. Can you explain this answer?

Question

Answer

Find the LCM of 21,36 and 66 search for options perfectly divisible by the LCM of 21,36 and 66.

Answer

You can solve by checking divisibility of 11 and 7By multiple of 11 option b and c can be eliminated.And by multiple of 7 option c also EliminatedRemain option A is divisible by 11 and 7 both.

Answer

Problem Analysis:
We are given three numbers: 21, 36, and 66. We need to find the least perfect square that is divisible by all three of these numbers.

Concept:
To find the least perfect square that is divisible by all three numbers, we need to find the least common multiple (LCM) of these three numbers and then find the smallest perfect square greater than or equal to the LCM.

Solution:
Let's find the LCM of 21, 36, and 66.

Finding the LCM:
- Prime factorization of 21: 21 = 3 × 7
- Prime factorization of 36: 36 = 2 × 2 × 3 × 3 = 2^2 × 3^2
- Prime factorization of 66: 66 = 2 × 3 × 11

To find the LCM, we take the highest exponent of each prime factor:
LCM = 2^2 × 3^2 × 7 × 11 = 2772

Finding the least perfect square greater than or equal to the LCM:
Now we need to find the least perfect square greater than or equal to 2772. We can do this by taking the square root of the LCM and rounding it up to the nearest integer.

√2772 ≈ 52.60

Rounding up, we get 53.

The least perfect square greater than or equal to 2772 is 53^2 = 2809.

Verifying the divisibility:
Now, we need to check if 2809 is divisible by 21, 36, and 66.

2809 ÷ 21 = 133 remainder 16
2809 ÷ 36 = 78 remainder 1
2809 ÷ 66 = 42 remainder 17

Since there are remainders in each division, 2809 is not divisible by any of the three numbers.

Finding the next perfect square:
We need to find the next perfect square greater than 2809 that is divisible by all three numbers.

Taking the square root of 2809, we get √2809 ≈ 53.00

Rounding up, we get 54.

The next perfect square is 54^2 = 2916.

Verifying the divisibility:
Now, we need to check if 2916 is divisible by 21, 36, and 66.

2916 ÷ 21 = 138 remainder 18
2916 ÷ 36 = 81 remainder 0
2916 ÷ 66 = 44 remainder 12

Since 2916 is divisible by all three numbers without any remainders, it is the least perfect square that is divisible by 21, 36, and 66.

Therefore, the correct answer is option A) 213444.

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Discussion on Perfect Square - Number Theory, Quantitative Aptitude?

how perfect square always odd no. Related: Perfect Square?

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