Find the smallest perfect square number divisible by `24`, `30` and `60`.

Options

A. `3600`
B. `3200`
C. `1600`
D. `2500`

Answer

To find the smallest perfect square number that is divisible by 24, 30, and 60, we will follow these steps:

### Step 1: Find the LCM of 24, 30, and 60
To find the least common multiple (LCM), we first find the prime factorization of each number:

- **24**: \( 24 = 2^3 \times 3^1 \)
- **30**: \( 30 = 2^1 \times 3^1 \times 5^1 \)
- **60**: \( 60 = 2^2 \times 3^1 \times 5^1 \)

Now, we take the highest power of each prime factor:

- For \(2\): the highest power is \(2^3\) (from 24)
- For \(3\): the highest power is \(3^1\) (from all three)
- For \(5\): the highest power is \(5^1\) (from 30 and 60)

So, the LCM is:
\[ \text{LCM} = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 \]

Calculating this:
\[ 8 \times 3 = 24 \]
\[ 24 \times 5 = 120 \]

Thus, the LCM of 24, 30, and 60 is **120**.

### Step 2: Check if the LCM is a Perfect Square
To determine if 120 is a perfect square, we look at its prime factorization:
\[ 120 = 2^3 \times 3^1 \times 5^1 \]

For a number to be a perfect square, all the exponents in its prime factorization must be even. Here:
- The exponent of \(2\) is \(3\) (odd)
- The exponent of \(3\) is \(1\) (odd)
- The exponent of \(5\) is \(1\) (odd)

Since not all exponents are even, **120 is not a perfect square**.

### Step 3: Make the LCM a Perfect Square
To make the LCM a perfect square, we need to adjust the exponents to the nearest even numbers:
- For \(2^3\), we need one more \(2\) to make it \(2^4\).
- For \(3^1\), we need one more \(3\) to make it \(3^2\).
- For \(5^1\), we need one more \(5\) to make it \(5^2\).

Thus, we need to multiply \(120\) by \(2^1 \times 3^1 \times 5^1\):
\[ 120 \times 2 \times 3 \times 5 \]

Calculating this:
\[ 120 \times 2 = 240 \]
\[ 240 \times 3 = 720 \]
\[ 720 \times 5 = 3600 \]

### Conclusion
The smallest perfect square that is divisible by 24, 30, and 60 is **3600**.

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