Final answer:
The contrapositive of the statement 'if 5n + 3 is even, then n is odd' is proven by assuming n is even and showing that 5n + 3 is then odd, thereby confirming the original statement.
Step-by-step explanation:
To prove the statement by contrapositive: 'For every integer n, if 5n + 3 is even, then n is odd', we must first understand what the contrapositive of a statement is. The contrapositive of a conditional statement 'if p, then q' is 'if not q, then not p'. In this case, the contrapositive would be 'if n is not odd (meaning n is even), then 5n + 3 is not even (meaning 5n + 3 is odd)'.
Now, if n is even, it can be expressed as n = 2k for some integer k. Substituting into 5n + 3, we get 5(2k) + 3 which simplifies to 10k + 3. Here, 10k is obviously even, but 10k + 3 is odd since an even number plus 3 results in an odd number. Thus, if n is even, 5n + 3 is odd, which confirms the contrapositive.
By proving the contrapositive, we have proven the original statement: If 5n + 3 is even, then n must be odd.