4.6: Section 6-Exercises

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Use the method of direct proof to prove the following statements.

Exercise \(\PageIndex{1}\)

If x is an even integer, then \(x^2\) is even.

Exercise \(\PageIndex{2}\)

If x is an odd integer, then \(x^3\) is odd.

Exercise \(\PageIndex{3}\)

If a is an odd integer, then \(a^2+3a+5\) is odd.

Exercise \(\PageIndex{4}\)

Suppose \(x,y \in \mathbb{Z}\). If x and y are odd, then xy is odd.

Exercise \(\PageIndex{5}\)

Suppose \(x, y \in \mathbb{Z}\). If x is even, then xy is even.

Exercise \(\PageIndex{6}\)

Suppose \(a, b, c \in \mathbb{Z}\). If \(a|b\) and \(a|c\), then \(a|(b+c)\).

Exercise \(\PageIndex{7}\)

Suppose \(a, b \in \mathbb{Z}\). If \(a|b\), then \(a^{2}|b^{2}\).

Exercise \(\PageIndex{8}\)

Suppose a is an integer. If \(5|2a\), then \(5|a\).

Exercise \(\PageIndex{9}\)

Suppose a is an integer. If \(7|4a\), then \(7|a\).

Exercise \(\PageIndex{10}\)

Suppose a and b are integers. If \(a|b\), then \(a|(3b^3-b2+5b)\).

Exercise \(\PageIndex{11}\)

Suppose \(a, b, c, d \in \mathbb{Z}\). If \(a|b\) and \(c|d\), then \(ac|bd\).

Exercise \(\PageIndex{12}\)

If \(x \in \mathbb{R}\) and \(0<x<4\), then \(\frac{4}{x(4−x)} \ge 1\).

Exercise \(\PageIndex{13}\)

Suppose \(x, y \in \mathbb{R}\). If \(x^2+5y = y^2+5x\), then \(x=y\) or \(x+y = 5\).

Exercise \(\PageIndex{14}\)

If \(n \in \mathbb{Z}\), then \(5n^2+3n+7\) is odd. (Trycases.)

Exercise \(\PageIndex{15}\)

If \(n \in \mathbb{Z}\), then \(n^2+3n+4\) is even. (Trycases.)

Exercise \(\PageIndex{16}\)

If two integers have the same parity, then their sum is even. (Try cases.)

Exercise \(\PageIndex{17}\)

If two integers have opposite parity, then their product is even.

Exercise \(\PageIndex{18}\)

Suppose x and y are positive real numbers. If \(x < y\), then \(x^{2} < y^{2}\).

Exercise \(\PageIndex{19}\)

Suppose a, b and c are integers. If \(a^{2}|b\) and \(b^{3}|c\), then \(a^6|c\).

Exercise \(\PageIndex{20}\)

If a is an integer and \(a^{2}|a\), then \(a \in \{−1, 0, 1\}\).

Exercise \(\PageIndex{21}\)

If p is prime and k is an integer for which \(0<k<p\), then p divides \({p \choose k}\).

Exercise \(\PageIndex{22}\)

If \(n \in \mathbb{N}\), then \(n^{2} = 2{n \choose 2} + {n \choose 1}\). (You may need a separate case for \(n = 1\).)

Exercise \(\PageIndex{23}\)

If \(n \in \mathbb{N}\), then \({2n \choose n}\) is even.

Exercise \(\PageIndex{24}\)

If \(n \in \mathbb{N}\) and \(n \ge 2\), then the numbers \(n!+2, n!+3, n!+4, n!+5, \cdots , n!+n\) are all composite. (Thus for any \(n \ge 2\), one can find \(n - 1\) consecutive composite numbers. This means there are arbitrarily large “gaps” between prime numbers.)

Exercise \(\PageIndex{25}\)

If \(a, b, c \in \mathbb{N}\) and \(c \le b \le a\), then \({a \choose b}{a \choose c} = {a \choose b-c}{a-b+c \choose c}\).

Exercise \(\PageIndex{26}\)

Every odd integer is a difference of two squares. (Example \(7 = 4^{2}-3^{2}\), etc.)

Exercise \(\PageIndex{27}\)

Suppose \(a, b \in \mathbb{N}\). If \(gcd(a, b) > 1\), then \(b|a\) or b is not prime.

Exercise \(\PageIndex{28}\)

Let \(a, b, c \in \mathbb{Z}\). Suppose a and b are not both zero, and \(c \ne 0\). Prove that \(c \cdot gcd(a,b) \le gcd(ca, cb)\).