Let P(n) be the statement `n(n + 1) (n + 2)` is an integral multiple of 12, then which of the following is not true?

Options

A. (a) P(3)
B. (b) P(4)
C. (c) P(5)
D. (d) P(6)

Answer

To solve the problem, we need to determine which of the statements \( P(n) \) is not true, where \( P(n) \) states that \( n(n + 1)(n + 2) \) is an integral multiple of 12. We will check each option one by one.

### Step-by-step Solution:

1. **Understanding the Statement**:
We need to verify if \( n(n + 1)(n + 2) \) is a multiple of 12 for the given values of \( n \).

2. **Option A: \( P(3) \)**:
- Calculate \( P(3) = 3 \times (3 + 1) \times (3 + 2) = 3 \times 4 \times 5 \).
- Calculate:
\[ 3 \times 4 = 12 \]
\[ 12 \times 5 = 60 \]
- Check if 60 is a multiple of 12:
\[ 60 \div 12 = 5 \quad (\text{which is an integer}) \]
- Conclusion: \( P(3) \) is true.

3. **Option B: \( P(4) \)**:
- Calculate \( P(4) = 4 \times (4 + 1) \times (4 + 2) = 4 \times 5 \times 6 \).
- Calculate:
\[ 4 \times 5 = 20 \]
\[ 20 \times 6 = 120 \]
- Check if 120 is a multiple of 12:
\[ 120 \div 12 = 10 \quad (\text{which is an integer}) \]
- Conclusion: \( P(4) \) is true.

4. **Option C: \( P(5) \)**:
- Calculate \( P(5) = 5 \times (5 + 1) \times (5 + 2) = 5 \times 6 \times 7 \).
- Calculate:
\[ 5 \times 6 = 30 \]
\[ 30 \times 7 = 210 \]
- Check if 210 is a multiple of 12:
\[ 210 \div 12 = 17.5 \quad (\text{not an integer}) \]
- Conclusion: \( P(5) \) is not true.

5. **Option D: \( P(6) \)**:
- Calculate \( P(6) = 6 \times (6 + 1) \times (6 + 2) = 6 \times 7 \times 8 \).
- Calculate:
\[ 6 \times 7 = 42 \]
\[ 42 \times 8 = 336 \]
- Check if 336 is a multiple of 12:
\[ 336 \div 12 = 28 \quad (\text{which is an integer}) \]
- Conclusion: \( P(6) \) is true.

### Final Conclusion:
The only option that is not true is \( P(5) \). Therefore, the answer is **Option C: \( P(5) \)**.