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# 7.5: Exercise

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Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters 4–7.

Exercise $$\PageIndex{1}$$

Suppose $$x \in \mathbb{Z}$$. Then x is even if and only if $$3x+5$$ is odd.

Exercise $$\PageIndex{2}$$

Suppose $$x \in \mathbb{Z}$$. Then x is odd if and only if $$3x+6$$ is odd.

Exercise $$\PageIndex{3}$$

Given an integer a, then $$a^3+a^2+a$$ is even if and only if a is even.

Exercise $$\PageIndex{4}$$

Given an integer a, then $$a^2+4a+5$$ is odd if and only if a is even.

Exercise $$\PageIndex{5}$$

An integer a is odd if and only if $$a^3$$ is odd.

Exercise $$\PageIndex{6}$$

Suppose $$x, y \in \mathbb{R}$$.Then $$x^3+x^{2}y = y^2+xy$$ if and only if $$y = x^2$$ or $$y = -x$$.

Exercise $$\PageIndex{7}$$

Suppose $$x, y \in \mathbb{R}$$. Then $$(x+y)^2 = x^2+y^2$$ if and only if $$x = 0$$ or $$y = 0$$.

Exercise $$\PageIndex{8}$$

Suppose $$a, b \in \mathbb{Z}$$. Prove that $$a \equiv b \pmod{10}$$ if and only if $$a \not\equiv b \pmod{2}$$ and $$a \not\equiv b \pmod{5}$$.

Exercise $$\PageIndex{9}$$

Suppose $$a \in \mathbb{Z}$$. Prove that $$14|a$$ if and only if $$7|a$$ and $$2|a$$.

Exercise $$\PageIndex{10}$$

If $$a \in \mathbb{Z}$$, then $$a^3 \equiv a \pmod{3}$$.

Exercise $$\PageIndex{11}$$

Suppose $$a, b \in \mathbb{Z}$$. Prove that $$(a-3)b^2$$ is even if and only if a is odd or b is even.

Exercise $$\PageIndex{12}$$

There exists a positive real number x for which $$x^2 < \sqrt{x}$$.

Exercise $$\PageIndex{13}$$

Suppose $$a, b \in \mathbb{Z}$$. If $$a+b$$ is odd, then $$a^2+b^2$$ is odd.

Exercise $$\PageIndex{14}$$

Suppose $$a \in \mathbb{Z}$$. Then $$a^2|a$$ if and only if $$a \in \{-1,0,1\}$$.

Exercise $$\PageIndex{15}$$

Suppose $$(a, b \in \mathbb{Z}$$. Prove that $$a+b$$ is even if and only if a and b have the same parity.

Exercise $$\PageIndex{16}$$

Suppose $$a,b \in \mathbb{Z}$$. If ab is odd, then $$a^2+b^2$$ is even.

Exercise $$\PageIndex{17}$$

There is a prime number between 90 and 100.

Exercise $$\PageIndex{18}$$

There is a set X for which $$\mathbb{N} \in X$$ and $$\mathbb{N} \subseteq X$$.

Exercise $$\PageIndex{19}$$

If $$n \in \mathbb{N}$$, then $$2^0+2^1+2^2+2^3+2^4+ \cdots +2^n = 2^{n+1}-1$$.

Exercise $$\PageIndex{20}$$

There exists an $$n \in \mathbb{N}$$ for which $$11|(2n-1)$$.

Exercise $$\PageIndex{21}$$

Every real solution of $$x^3+x+3 = 0$$ is irrational.

Exercise $$\PageIndex{22}$$

If $$n \in \mathbb{Z}$$, then $$4|n^2$$ or $$4|(n^2-1)$$.

Exercise $$\PageIndex{23}$$

Suppose a, b and c are integers. If $$a|b$$ and $$a|(b^2-c)$$, then $$a|c$$.

Exercise $$\PageIndex{24}$$

If $$a \in \mathbb{Z}$$, then $$4 \nmid (a^2-3)$$.

Exercise $$\PageIndex{25}$$

If $$p > 1$$ is an integer and $$n \nmid p$$ for each integer n for which $$2 \le n \le \sqrt{p}$$, then p is prime.

Exercise $$\PageIndex{26}$$

The product of any n consecutive positive integers is divisible by n!.

Exercise $$\PageIndex{27}$$

Suppose $$a, b \in \mathbb{Z}$$. If $$a^2+b^2$$ is a perfect square, then a and b are not both odd.

Exercise $$\PageIndex{28}$$

Prove the division algorithm: If $$a, b \in \mathbb{N}$$, there exist unique integers q, r for which $$a = bq+r$$, and $$0 \le r < b$$. (A proof of existence is given in Section 1.9, but uniqueness needs to be established too.)

Exercise $$\PageIndex{29}$$

If $$a|bc$$ and $$gcd(a, b) = 1$$, then $$a|c$$. (Suggestion: Use the proposition on page 152.)

Exercise $$\PageIndex{30}$$

Suppose $$a, b, p \in \mathbb{Z}$$ and p is prime. Prove that if $$p|ab$$ then $$p|a$$ or $$p|b$$. (Suggestion: Use the proposition on page 152.)

Exercise $$\PageIndex{31}$$

If $$n \in \mathbb{Z}$$, then $$gcd(n, n+1) = 1$$.

Exercise $$\PageIndex{32}$$

If $$n \in \mathbb{Z}$$, then $$gcd(n, n+2) \in \{1, 2\}$$.

Exercise $$\PageIndex{33}$$

If $$n \in \mathbb{Z}$$, then $$gcd(2n+1, 4n^2+1) = 1$$.

Exercise $$\PageIndex{34}$$

If $$gcd(a, c) = gcd(b, c) = 1$$, then $$gcd(ab, c) = 1$$. (Suggestion: Use the proposition on page 152.)

Exercise $$\PageIndex{35}$$

Suppose $$a, b \in \mathbb{N}$$. Then $$a = gcd(a,b)$$ if and only if $$a|b$$.

Exercise $$\PageIndex{36}$$

Suppose $$a, b \in \mathbb{N}$$. Then $$a = lcm(a,b)$$ if and only if $$b|a$$.