The least perfect square divisible by 24, 30, and 60 is 3600.

Quick Scoop: Key Idea

To be divisible by 24, 30, and 60, a number must be a multiple of their LCM (Least Common Multiple).
Then, to also be a perfect square , every prime factor in that LCM must appear with an even exponent.

Step 1: Find LCM of 24, 30, 60

Prime factorization:

  • 24=23×324=2^3\times 324=23×3
  • 30=2×3×530=2\times 3\times 530=2×3×5
  • 60=22×3×560=2^2\times 3\times 560=22×3×5

Take the highest powers of each prime:

  • For 2: highest power is 232^323
  • For 3: highest power is 313^131
  • For 5: highest power is 515^151

So,

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

Any number divisible by 24, 30, and 60 must be a multiple of 120.

Step 2: Make this into a perfect square

LCM in prime form:

120=23×31×51120=2^3\times 3^1\times 5^1120=23×31×51

For a perfect square , all exponents must be even. Currently:

  • Exponent of 2 is 3 → needs 1 more 2 to become 242^424
  • Exponent of 3 is 1 → needs 1 more 3 to become 323^232
  • Exponent of 5 is 1 → needs 1 more 5 to become 525^252

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

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

Now check the prime factorization:

3600=120×30=(23×3×5)×(2×3×5)=23+1×31+1×51+1=24×32×523600=120\times 30=(2^3\times 3\times 5)\times (2\times 3\times 5) =2^{3+1}\times 3^{1+1}\times 5^{1+1} =2^4\times 3^2\times 5^23600=120×30=(23×3×5)×(2×3×5)=23+1×31+1×51+1=24×32×52

All exponents are even, so 3600 is a perfect square.
Also:

  • 3600÷24=1503600\div 24=1503600÷24=150
  • 3600÷30=1203600\div 30=1203600÷30=120
  • 3600÷60=603600\div 60=603600÷60=60

So 3600 is divisible by 24, 30, and 60.

Final Answer

The least perfect square which is divisible by 24, 30, and 60 is 3600.

Information gathered from public forums or data available on the internet and portrayed here.