A number P is divisible by 5,4,8 and 9. If p is a 3-digit number, then find all the possible values for P.

Explanation

To solve the problem of finding all possible three-digit values for \( P \) that are divisible by 5, 4, 8, and 9, we can follow these steps:

### Step 1: Find the Least Common Multiple (LCM)
To determine the values of \( P \), we first need to find the LCM of the numbers 5, 4, 8, and 9. The LCM is the smallest number that is divisible by all of these numbers.

1. **Prime Factorization**:
- \( 5 = 5^1 \)
- \( 4 = 2^2 \)
- \( 8 = 2^3 \)
- \( 9 = 3^2 \)

2. **Taking the highest power of each prime**:
- For \( 2 \): the highest power is \( 2^3 \) (from 8)
- For \( 3 \): the highest power is \( 3^2 \) (from 9)
- For \( 5 \): the highest power is \( 5^1 \) (from 5)

3. **Calculating the LCM**:
\[ \text{LCM} = 2^3 \times 3^2 \times 5^1 = 8 \times 9 \times 5 \]
\[ = 72 \times 5 = 360 \]

### Step 2: Finding the Multiples of the LCM
Now that we have the LCM, which is 360, we need to find the multiples of 360 that are three-digit numbers.

1. **Finding the multiples**:
- The first multiple is \( 360 \times 1 = 360 \) (which is a three-digit number).
- The second multiple is \( 360 \times 2 = 720 \) (which is also a three-digit number).
- The third multiple is \( 360 \times 3 = 1080 \) (which is a four-digit number and not valid).

### Step 3: Listing the Possible Values
Thus, the possible three-digit values for \( P \) are:
- \( 360 \)
- \( 720 \)

### Final Answer
The possible values for \( P \) are **360 and 720**.

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