Proof by contraposition involves proving the contrapositive statement of a conditional statement. The contrapositive of "if n is an integer and n^3 + 5 is odd, then n is even" is "if n is an odd integer, then n^3 + 5 is even."
Proof:
Assume n is an odd integer, which means that n can be written as 2k + 1 for some integer k.
Substituting this expression for n in the expression n^3 + 5, we get:
n^3 + 5 = (2k + 1)^3 + 5
n^3 + 5 = 8k^3 + 12k^2 + 6k + 6
Since 8k^3 + 12k^2 + 6k is even for any integer k, we can conclude that n^3 + 5 is even.
Therefore, the contrapositive statement "if n is an odd integer, then n^3 + 5 is even" is true, which implies that the original statement "if n is an integer and n^3 + 5 is odd, then n is even" is also true.
Hence, we have proved that if n^3 + 5 is odd, then n is even using proof by contraposition.