Which of the following statements is/are false?
p : 137 is an integer or rational number
q : Birds has wings or colour of sky is blue
r : One of every three consecutive integer is divisible by 3 or product of every two consecutive integer is odd

Options

A. p only
B. q only
C. p and q only
D. p, q, r

Answer

To determine which of the statements is/are false, let's analyze each statement one by one.

### Statement p:
"137 is an integer or rational number."

1. **Analysis**:
- 137 is a whole number, which means it is an integer.
- All integers are also rational numbers because they can be expressed as a fraction (e.g., \( \frac{137}{1} \)).

2. **Conclusion**:
- This statement is **true**.

### Statement q:
"Birds have wings or the color of the sky is blue."

1. **Analysis**:
- The first part of the statement, "Birds have wings," is true because all birds possess wings.
- The second part, "the color of the sky is blue," is generally true during a clear day but can be false during other times (e.g., at night or when it’s cloudy).
- However, since the statement uses "or," it only requires one part to be true for the entire statement to be true.

2. **Conclusion**:
- This statement is **true**.

### Statement r:
"One of every three consecutive integers is divisible by 3 or the product of every two consecutive integers is odd."

1. **Analysis**:
- Let's break this down into two parts:
- **Part 1**: "One of every three consecutive integers is divisible by 3."
- Consider three consecutive integers: \( n, n+1, n+2 \).
- Among these, one will always be divisible by 3. For example:
- If \( n = 3k \) (where \( k \) is an integer), then \( n \) is divisible by 3.
- If \( n = 3k + 1 \), then \( n + 2 \) is divisible by 3.
- If \( n = 3k + 2 \), then \( n + 1 \) is divisible by 3.
- Therefore, this part is **true**.

- **Part 2**: "The product of every two consecutive integers is odd."
- For any two consecutive integers \( n \) and \( n+1 \), one of them will always be even (e.g., if \( n \) is even, \( n+1 \) is odd, and vice versa).
- The product of an even number and any other integer (even or odd) is always even.
- Therefore, this part is **false**.

2. **Conclusion**:
- Since the first part of statement r is true but the second part is false, the overall statement is **false**.

### Final Conclusion:
- Statement p is true.
- Statement q is true.
- Statement r is false.

Thus, the answer is that **statement r is false**.

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