Let n ∈ Z.
(1) Give a direct proof of the following: If n is odd, then 4 | n
2 − 1.
(2) State the contrapositive of the implication in (1).
(3) Give a direct proof of the following: If n is even, then 4 doesn’t divide n
2 − 1.
(4) State the contrapositive of the implication in (3).
(5) State the conjunction of the implications in (1) and (3) using “if and only if”.
+23246 1) Direct proof of: If n is odd, then 4 divides (n2 - 1).
If n is odd, then there exists an integer k such that 2k + 1 = n.
Then: n2 - 1 = (2k + 1)2 - 1 = (4k2 + 4k + 1) - 1 = 4k2 + 4k = 4(k2 + k)
and 4(k2 + k) / 4 = k2 + k
Also; since k is an integer, k2 + k is an integer; therefore the proof is complete.
2) Contrapositive of: "if n is odd, then 4 divides (n2 - 1)" is "if 4 divides (n2 - 1), then n is odd".
3) Direct proof of: If n is even, then 4 doesn't divide n2 -1.
If n is even, then there exists an integer k such that 2k = n.
Then: n2 - 1 = (2k)2 - 1 = 4k2 - 1
and (4k2 - 1) / 4 = 4k2/4 - 1/4 = k2 - 1/4
estblishing a remainder of 1/4 (or, as an integer, either -3 or +3).
4) Contrapositive of: "if n is even, then 4 doesn't divide n2 - 1" is " if 4 doesn't divide n2 - 1, then n is even".
5) 4 divides n2 - 1 if and only if n is odd.
