
# 2 E: Exercises

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Exercise $$\PageIndex{1}$$:

Determine whether or not each of the following binary relations $$R$$ on the given set $$A$$ is reflexive, symmetric,
antisymmetric, or transitive. If a relation has a certain property, prove this is so; otherwise, provide a counterexample
to show that it does not. If $$R$$ is an equivalence relation, describe the equivalence classes of $$A$$.

1. Let $$S=\{0,1,2,3,4,5,6,7,8,9 \}.$$ Define a relation $$R$$ on $$A= S \times S$$ by $$(a,b)\, R\, (c,d)$$ if and only if $$10 a +b \leq 10 c+ d.$$

2. Let $$A= \mathbb{Z} \backslash \{0\}.$$ Define a relation $$R$$ on $$A,$$ by $$a \,R\, b$$ if and only if $$ab>0.$$

3. Define a relation $$R$$ on $$A={\mathbb Z}$$ by $$a \,R\, b$$ if and only if $$4 \mid (3a+b).$$
4. Define a relation $$R$$ on $$A={\mathbb Z}$$ by $$a \, R \, b$$ if and only if $$3 \mid (a^2-b^2).$$

5. Let $$A=\mathbb{R}$$, If $$a.b \in \mathbb{R}$$, define $$a \, R \, b$$ if and only if $$a-b \in \mathbb{Z}.$$

6. Define a relation $$R$$ on the set $${\mathbb Z} \times{\mathbb Z}$$ by: $$(a,b)\, R \,(c,d) \mbox{ if and only if } ac=bd.$$

7. Define a relation $$R$$ on $${\mathbb Z}$$ by: $$a R b$$ if and only if $$5 \mid 2a+3b.$$

8. Define a relation $$R$$ on $${\mathbb Z}$$ by: $$a R b$$ if and only if $$2 \mid a^2+b.$$

Exercise $$\PageIndex{2}$$:

Let $$a, b,c, d \in \bf Z_+.$$

1. If $$a|b$$ and $$a|c$$ , is it necessarily true that $$a|(b + c)?$$
2. If $$a|(b + c)$$, is it necessarily true that $$a|b$$ and $$a|c$$?
3. If $$a|bc$$, is it necessarily true that $$a|b$$ and $$a|c$$?
4. If $$(a+b)|c$$, is it necessarily true that $$a|c$$ and $$b|c$$?
5. If $$a|c$$ and $$b|c$$ , is it necessarily true that $$(a+b)|c?$$
6. If $$a^3|b^4$$, is it necessarily true that $$a|b.$$
7. If $$a|b$$ , is it necessarily true that $$a^3 \mid b^5?$$
8. If $$c|a$$ and $$d|b$$, is it necessarily true that $$cd|ab$$.

Exercise $$\PageIndex{3}$$:

1. Find all possible values for the missing digit if $$12345X51234$$ is divisible by $$3.$$
2. Using divisibility tests, check if the number $$355581$$ is divisible by $$7$$
3. Using divisibility tests, check if the number $$824112284$$ is divisible by $$5, 4,$$ and $$8.$$

Exercise $$\PageIndex{4}$$:

Let $$a$$ and $$b$$ be positive integers such that $$7|(a+2b-2)$$ and $$7|(b-9).$$ Prove that $$7|(a+b).$$

Exercise $$\PageIndex{5}$$:

1. In a $$113$$-digit multiple of $$13$$, the first $$56$$ digits are all $$5$$s and the last $$56$$ digits are all $$8$$s. What is the middle digit?
2. In a $$113$$-digit multiple of $$7$$, the first $$56$$ digits are all $$8$$s and the last $$56$$ digits are all $$1$$s. What is the middle digit?

Exercise $$\PageIndex{6}$$:

Prove the statements that are true and give counterexamples to disprove those that are false.
Let $$a,b$$ and $$c$$ be integers.

1. If $$a|b$$ then $$b|a$$.
2. If $$a|bc$$ then $$a|b$$ and $$a|c.$$
3. If $$a|b$$ and $$a|c$$ then $$a|bc$$.
4. If $$a|b$$ and $$a|c$$ then $$a|(b+c)$$ and $$a|(b-c)$$.
5. If $$a|(b+c)$$ and $$a|(b-c)$$ then $$a|b$$ and $$a|c$$.
6. If $$a|b$$ and $$a|c$$ then $$a|(b+c)$$ and $$a|(2b+c)$$.
7. If $$a|(b+c)$$ and $$a|(2b+c)$$ then $$a|b$$ and $$a|c$$

Exercise $$\PageIndex{7}$$

Prove that for all integers $$n\geq 1, \,5^{2n}-2^{5n}$$ is divisible by $$7.$$