Find the least square number which is exactly divisible by each of the numbes `8,12,15 and 20.`

Answer

To find the least square number that is exactly divisible by the numbers 8, 12, 15, and 20, we will follow these steps:

### Step 1: Find the LCM of the numbers
We need to find the Least Common Multiple (LCM) of the numbers 8, 12, 15, and 20.

1. **Prime Factorization**:
- 8 = 2^3
- 12 = 2^2 × 3^1
- 15 = 3^1 × 5^1
- 20 = 2^2 × 5^1

2. **Identify the highest powers of each prime factor**:
- For 2: highest power is 2^3 (from 8)
- For 3: highest power is 3^1 (from 12 and 15)
- For 5: highest power is 5^1 (from 15 and 20)

3. **Calculate the LCM**:
\[ \text{LCM} = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 \]
\[ = 24 \times 5 = 120 \]

### Step 2: Check the LCM
We have found that the LCM of 8, 12, 15, and 20 is 120. This means 120 is divisible by all these numbers.

### Step 3: Make the LCM a perfect square
Next, we need to ensure that the LCM (120) is a perfect square. To do this, we will look at the prime factorization of 120 and adjust it to make all the powers even.

- The prime factorization of 120 is:
\[ 120 = 2^3 \times 3^1 \times 5^1 \]

- To make this a perfect square:
- For 2^3, we need one more 2 to make it 2^4 (even).
- For 3^1, we need one more 3 to make it 3^2 (even).
- For 5^1, we need one more 5 to make it 5^2 (even).

### Step 4: Multiply to get the perfect square
Now we will multiply the LCM by the necessary factors to make all the powers even:
\[ \text{Perfect square} = 120 \times 2^1 \times 3^1 \times 5^1 \]
Calculating this:
\[ = 120 \times 2 \times 3 \times 5 \]
Calculating step by step:
1. \(120 \times 2 = 240\)
2. \(240 \times 3 = 720\)
3. \(720 \times 5 = 3600\)

### Conclusion
The least square number which is exactly divisible by 8, 12, 15, and 20 is **3600**.