5.4: Section 4-

A. Prove the following statements with contrapositive proof. (In each case, think about how a direct proof would work. In most cases contrapositive is easier.)

Exercise \(\PageIndex{1}\)

Suppose \(n \in \mathbb{Z}\). If \(n^2\) is even, then n is even.

Exercise \(\PageIndex{2}\)

Suppose \(n \in \mathbb{Z}\). If \(n^2\) is odd, then n is odd.

Exercise \(\PageIndex{3}\)

Suppose \(a, b \in \mathbb{Z}\). If \(a^{2}(b^2-2b)\) is odd, then a and b are odd.

Exercise \(\PageIndex{4}\)

Suppose \(a, b, c \in \mathbb{Z}\). If a does not divide bc, then a does not divide b.

Exercise \(\PageIndex{5}\)

Suppose \(x \in \mathbb{R}\). If \(x^2+5x < 0\) then \(x < 0\).

Exercise \(\PageIndex{6}\)

Suppose \(x \in \mathbb{R}\). If \(x^3-x > 0\) then \(x > -1\).

Exercise \(\PageIndex{7}\)

Suppose \(a, b \in \mathbb{Z}\). If both ab and \(a+b\) are even, then both a and b are even.

Exercise \(\PageIndex{8}\)

Suppose \(x \in \mathbb{R}\). If \(x^5-4x^4+3x^3-x^2+3x-4 \ge 0\), then \(x \ge 0\).

Exercise \(\PageIndex{9}\)

Suppose \(n \in \mathbb{Z}\). If \(3�� \nmid n^2\), then \(3�� \nmid n\).

Exercise \(\PageIndex{10}\)

Suppose \(x, y, z \in \mathbb{Z}\) and \(x \ne 0\). If \(x \nmid ��yz\), then \(x \nmid ��y\) and \(x \nmid ��z\).

Exercise \(\PageIndex{11}\)

Suppose \(x, y \in \mathbb{Z}\). If \(x^{2}(y+3)\) is even, then x is even or y is odd.

Exercise \(\PageIndex{12}\)

Suppose \(a \in \mathbb{Z}\). If \(a^2\) is not divisible by 4, then a is odd.

Exercise \(\PageIndex{13}\)

Suppose \(x \in \mathbb{R}\). If \(x^5+7x^3+5x \ge x^4+x^2+8\), then \(x \ge 0\).

B. Prove the following statements using either direct or contrapositive proof.

Exercise \(\PageIndex{14}\)

If \(a, b \in \mathbb{Z}\) and a and b have the same parity, then \(3a+7\) and \(7b-4\) do not.

Exercise \(\PageIndex{15}\)

Suppose \(x \in \mathbb{Z}\). If \(x^3-1\) is even, then x is odd.

Exercise \(\PageIndex{16}\)

Suppose \(x, y \in \mathbb{Z}\). If \(x+y\) is even, then x and y have the same parity.

Exercise \(\PageIndex{17}\)

If n is odd, then \(8|(n^2-1)\).

Exercise \(\PageIndex{18}\)

If \(a, b \in \mathbb{Z}\), then \((a+b)^{3} \equiv a^3+b^3 \pmod{3}\).

Exercise \(\PageIndex{19}\)

Let \(a, b, c \in \mathbb{Z}\) and \(n \in \mathbb{N}\). If  \(a \equiv b \pmod{n}\) and \(a \equiv c \pmod{n}\), then \(c \equiv b \pmod{n}\).

Exercise \(\PageIndex{20}\)

If \(a \in \mathbb{Z}\) and \(a \equiv 1 \pmod{5}\), then \(a^2 \equiv 1 \pmod{5}\).

Exercise \(\PageIndex{21}\)

Let \(a, b \in \mathbb{Z}\) and \(n \in \mathbb{N}\). If \(a \equiv b \pmod{n}\), then \(a^3 \equiv b^3 \pmod{n}\).

 

 

 

 

 

14. Let a∈Z, n∈N. If a has remainder r when divided by n, then a≡r (mod n).

15. Leta,b∈Zandn∈N.Ifa≡b(modn),thena2≡ab(modn).

16. If a≡b (mod n) and c≡d (mod n), then ac≡bd (mod n).

17. Letn∈N.If2n−1isprime,thennisprime.

18. Ifn=2k−1fork∈N,theneveryentryinRownofPascal’sTriangleisodd.

19. If a≡0 (mod 4) or a≡1 (mod 4), then ��a2�� is even.

20. Ifn∈Z,then4��(n2−3).

21. Ifintegersaandbarenotbothzero,thengcd(a,b)=gcd(a−b,b).

22. Ifa≡b(modn),thengcd(a,n)=gcd(b,n).

23. Suppose the division algorithm applied to a and b yields a = qb + r. Prove gcd(a, b) = gcd(r, b).

24. If a ≡ b (mod n), then a and b have the same remainder when divided by n