Let `P(n) :n(n+1) (n+2)` is divisble by 3. What is P(3)?

Answer

To solve the problem, we need to evaluate \( P(3) \) where \( P(n) : n(n+1)(n+2) \) is divisible by 3.

### Step-by-Step Solution:

1. **Identify the expression for \( P(n) \)**:
\[ P(n) = n(n+1)(n+2) \]
This expression represents the product of three consecutive integers: \( n \), \( n+1 \), and \( n+2 \).

2. **Substitute \( n = 3 \)** into the expression:
\[ P(3) = 3(3+1)(3+2) \]

3. **Calculate \( P(3) \)**:
- First, calculate \( 3 + 1 \):
\[ 3 + 1 = 4 \]
- Next, calculate \( 3 + 2 \):
\[ 3 + 2 = 5 \]
- Now substitute these values back into the expression:
\[ P(3) = 3 \times 4 \times 5 \]

4. **Perform the multiplication**:
- Calculate \( 3 \times 4 \):
\[ 3 \times 4 = 12 \]
- Now multiply this result by 5:
\[ 12 \times 5 = 60 \]

5. **Conclusion**:
\[ P(3) = 60 \]
Since 60 is divisible by 3, we conclude that \( P(3) \) is indeed divisible by 3.

### Final Answer:
\[ P(3) = 60 \quad \text{and it is divisible by 3.} \]