The smallest square number divisible by 10, 16 and 24 is:

Options

A. 900
B. 1600
C. 2500
D. 3600

Answer

To find the smallest square number that is divisible by 10, 16, and 24, we can follow these steps:

### Step 1: Find the LCM of 10, 16, and 24
To find the least common multiple (LCM), we first need to determine the prime factorization of each number:

- **10** = 2 × 5
- **16** = 2^4
- **24** = 2^3 × 3

Next, we take the highest power of each prime factor that appears in the factorizations:

- For the prime factor **2**, the highest power is 2^4 (from 16).
- For the prime factor **3**, the highest power is 3^1 (from 24).
- For the prime factor **5**, the highest power is 5^1 (from 10).

Now we can calculate the LCM:

\[ \text{LCM} = 2^4 \times 3^1 \times 5^1 \]

### Step 2: Calculate the LCM
Calculating the LCM step by step:

1. Calculate \(2^4 = 16\)
2. Calculate \(3^1 = 3\)
3. Calculate \(5^1 = 5\)

Now multiply these together:

\[ \text{LCM} = 16 \times 3 \times 5 \]

Calculating this step by step:

- First, \(16 \times 3 = 48\)
- Then, \(48 \times 5 = 240\)

So, the LCM of 10, 16, and 24 is **240**.

### Step 3: Make the LCM a Perfect Square
Now, we need to find the smallest perfect square that is divisible by 240.

The prime factorization of 240 is:

\[ 240 = 2^4 \times 3^1 \times 5^1 \]

For a number to be a perfect square, all the exponents in its prime factorization must be even.

- The exponent of **2** is 4 (even).
- The exponent of **3** is 1 (odd), so we need to increase it to 2.
- The exponent of **5** is 1 (odd), so we need to increase it to 2.

Thus, we need to multiply 240 by \(3^1\) and \(5^1\) to make the exponents even:

\[ \text{Required number} = 3^1 \times 5^1 = 15 \]

### Step 4: Calculate the Smallest Perfect Square
Now we multiply the LCM by this required number:

\[ \text{Smallest perfect square} = 240 \times 15 \]

Calculating this:

1. \(240 \times 15 = 3600\)

Thus, the smallest square number divisible by 10, 16, and 24 is **3600**.

### Final Answer
The smallest square number divisible by 10, 16, and 24 is **3600**.

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