The Product of a Whole Number and Its Successor
When we consider a non-zero whole number \( n \) and its successor \( n + 1 \), we can analyze the product \( n(n + 1) \). This product can be examined for its properties such as whether it is even, odd, prime, or divisible by 3.
1. Is the Product Always Even?
- The product \( n(n + 1) \) is always even.
- This is because one of the two consecutive numbers, \( n \) or \( n + 1 \), must be even.
- An even number times any whole number results in an even product.
2. Is the Product Always Odd?
- The product \( n(n + 1) \) is never odd.
- Since one of the consecutive integers is always even, the product cannot be odd.
3. Is the Product Always Prime?
- The product \( n(n + 1) \) is not always prime.
- For example, if \( n = 2 \), then \( 2 \times 3 = 6 \), which is not a prime number.
- Prime numbers have exactly two distinct positive divisors, and the product of two whole numbers has more than two divisors.
4. Is the Product Always Divisible by 3?
- The product \( n(n + 1) \) is not always divisible by 3.
- Only one of every three consecutive numbers is divisible by 3. Thus, the product may or may not be divisible by 3 depending on the specific values of \( n \).
Conclusion
The product of a non-zero whole number and its successor is always an even number, while it can be odd, prime, or divisible by 3 depending on the specific numbers involved.