I. Two consecutive natural numbers are always co-prime.
II. If m and n are relatively prime, then m × n is always even.
Which of the statements given above is/are correct ?
To determine which statements are correct, we need to analyze each statement logically.
- Statement I: Two consecutive natural numbers are always co-prime.
- Consecutive natural numbers are two numbers that follow each other, such as (1, 2), (2, 3), (3, 4), etc.
- Co-prime numbers (or relatively prime numbers) are those numbers whose greatest common divisor (GCD) is 1.
- For example, take two consecutive numbers: 4 and 5. The factors of 4 are 1, 2, and 4, and the factors of 5 are 1 and 5. The only common factor between them is 1.
- This logic holds true for any pair of consecutive natural numbers.
- Statement II: If \( m \) and \( n \) are relatively prime, then \( m \times n \) is always even.
- Relatively prime numbers have a GCD of 1, meaning they do not have any common factors other than 1.
- For numbers \( m = 3 \) and \( n = 5 \), both are relatively prime but their product \( 3 \times 5 = 15 \), which is odd.
- This shows that the product \( m \times n \) can be odd if both \( m \) and \( n \) are odd numbers.
Based on the above analysis, the correct answer is: I only.
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