Shortcut Trick
Find the LCM of 12, 18, 24, and 27 which is 216.
Prime factorization of 216 = 23 × 33 = (22 × 2) × (32 × 3).
To make it a perfect square, we need to complete the pairs by multiplying by (2 × 3) = 6.
Smallest perfect square = 216 × 6 = 1296.
∴ The correct answer is 1296.
Alternate Method
Given: Numbers are 12, 18, 24, and 27.
Formula Used: LCM × Missing Factors = Smallest Perfect Square.
⇒ Prime factors of 12 = 22 × 3
⇒ Prime factors of 18 = 2 × 32
⇒ Prime factors of 24 = 23 × 3
⇒ Prime factors of 27 = 33
⇒ LCM = 23 × 33 = 8 × 27 = 216
⇒ To be a perfect square, each prime factor must have an even exponent.
⇒ Smallest multiple of 216 that is a square = 216 × 21 × 31
⇒ Required Number = 216 × 6 = 1296
∴ The correct answer is 1296.
Additional Information
Least Common Multiple (LCM)
The smallest number that is a multiple of two or more numbers. It is found by taking the highest power of each prime factor present in the numbers.
Perfect Square Property
For a number to be a perfect square, all the prime factors in its prime factorization must have even exponents (e.g., 22, 34, 56).
Finding Square from LCM
If LCM = ap × bq, then the smallest perfect square multiple is ap+x × bq+y where (p+x) and (q+y) are the smallest even integers ≥ p and q.