The least perfect square divisible by both 24 and 30 can be found by following these steps: Step 1: Find the LCM of 24 and 30

• Prime factorization of 24 = 23×312^3\times 3^123×31
• Prime factorization of 30 = 21×31×512^1\times 3^1\times 5^121×31×51

The LCM is the product of the highest powers of all primes involved:

LCM=23×31×51=8×3×5=120LCM=2^3\times 3^1\times 5^1=8\times 3\times 5=120LCM=23×31×51=8×3×5=120

Step 2: Check if the LCM is a perfect square

• 120=23×31×51120=2^3\times 3^1\times 5^1120=23×31×51
• The exponents are 3 (odd) for 2, 1 (odd) for 3, and 1 (odd) for 5.

Since all exponents must be even for a perfect square, 120 is not a perfect square. Step 3: Adjust the LCM to form the least perfect square divisible by 24 and 30 Make each exponent even by multiplying by the necessary primes:

• For 232^323, multiply by one more 2 to get 242^424
• For 313^131, multiply by one more 3 to get 323^232
• For 515^151, multiply by one more 5 to get 525^252

Thus, multiply 120 by 2×3×5=302\times 3\times 5=302×3×5=30:

120×30=3600120\times 30=3600120×30=3600

Step 4: Verify

• 3600=24×32×523600=2^4\times 3^2\times 5^23600=24×32×52, all exponents are even, so 3600 is a perfect square.
• 3600 is divisible by both 24 and 30.

Conclusion: The least perfect square divisible by both 24 and 30 is 3600