Final answer:
To prove the statement using proof by contraposition, we need to show that if n is not an even positive integer, then n² is not an even positive integer. Let's assume that n is not an even positive integer, which means n is either odd or not positive. If n is odd, we can write it as n = 2k + 1, where k is an integer.
Step-by-step explanation:
To prove the statement using proof by contraposition, we need to show that if n is not an even positive integer, then n² is not an even positive integer.
Let's assume that n is not an even positive integer, which means n is either odd or not positive.
If n is odd, we can write it as n = 2k + 1, where k is an integer.
Squaring both sides, we get n² = (2k + 1)² = 4k² + 4k + 1 = 2(2k² + 2k) + 1, which shows that n² is an odd number and not an even positive integer.
If n is not positive, it can be written as n = -m, where m is a positive integer.
Squaring both sides, we get n² = (-m)² = m², which is a positive integer but not an even positive integer.
Therefore, we have shown that if n is not an even positive integer, then n² is not an even positive integer.