Discussion
Consider the following statements:
1. The cube of any even number is always even.
2. The square of any odd number is always odd.
3. The product of two odd numbers is always odd.
How many of the statements given above are correct?
Answer: C
All three statements are correct.
Statement 1: an even number is 2k, and \((2k)^3 = 8k^3\), which is even.
Statement 2: an odd number is \(2k + 1\), and \((2k + 1)^2 = 4k^2 + 4k + 1\), which is odd.
Statement 3: \((2a + 1)(2b + 1) = 4ab + 2a + 2b + 1\), which is odd.
The general rule behind all three: a product is even if and only if at least one factor is even, so powers of even numbers stay even and products of odd numbers stay odd.
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