# 2.E: Exercises

- Page ID
- 7429

### Exercise \(\PageIndex{1}\): Equivalence relation

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 \(A\).

- Define a relation \(R\) on \(A={\mathbb Z}\) by \(a \,R\, b\) if and only if \( 4 \mid (3a+b).\)
- Define a relation \(R\) on \(A={\mathbb Z}\) by \(a \, R \, b\) if and only if \( 3 \mid (a^2-b^2).\)
- Let \(A=\mathbb{R}\), If \(a.b \in \mathbb{R}\), define \(a \, R \, b\) if and only if \( a-b \in \mathbb{Z}.\)
- Define a relation \(R\) on the set \({\mathbb Z} \times{\mathbb Z}\) by: \((a,b)\, R \,(c,d) \mbox{ if and only if } ac=bd.\)
- Define a relation \(R\) on \({\mathbb Z}\) by: \(a R b\) if and only if \( 5 \mid 2a+3b.\)
- Define a relation \(R\) on \({\mathbb Z}\) by: \(a R b\) if and only if \( 2 \mid a^2+b.\)

### Exercise \(\PageIndex{2}\): Divides

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

- If \(a|b\) and \(a|c\) , is it necessarily true that \(a|(b + c)?\)
- If \(a|(b + c)\), is it necessarily true that \(a|b\) and \(a|c\)?
- If \(a|bc\), is it necessarily true that \(a|b\) and \(a|c\)?
- If \((a+b)|c\), is it necessarily true that \(a|c\) and \(b|c\)?
- If \(a|c\) and \(b|c\) , is it necessarily true that \((a+b)|c?\)
- If \(a^3|b^4\), is it necessarily true that \(a|b.\)
- If \(a|b\) , is it necessarily true that \(a^3 \mid b^5?\)
- If \(c|a\) and \(d|b\), is it necessarily true that \(cd|ab\).

### Exercise \(\PageIndex{3}\): Divisibility Rules

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

**Answer:**-
1. An integer (q) is divisible by 3 iff i=0ndiis divisible by 3, where di is the ith digit of q. Thus, the sum of the digits of 12345X51234 must be divisible by 3.

Simplifying, 3 | (30 + X).

Since 3 | 30, 3 must divide X in order to satisfy 3 |12345X51234.

Thus \( X ∈ { 0, 3, 6, or 9}.\)

2.

An integer (q) is divisible by 7 iff 7 | (a - 2b), where b is the last digit of q and a is the remaining digits.

*Step 1*: a = 35558, b = 1.Thus, a - 2b = 35556.

*Step 2*: a = 3555, b = 6.Thus, a - 2b = 3543.

Note at this point it is clear that 7 ∤ 355581, since 7 ∤ 3543. However, for practice, we will continue on.

*Step 3*: a = 354, b = 3.Thus, a - 2b = 348.

*Step 4*: a = 34, b = 8.Thus a - 2b = 18.

Since 7 ∤ 18, 7 ∤ 355581.

3.

**Solution:**An integer (q), is divisible by 5 iff its last digits ∈ {0, 5}.

Since 4 ∉ {0, 5}, 5 ∤ 824112284.

**Solution:**An integer (q) is divisible by 8 iff q’s last three digits are divisible by 8.

Since 8 ∤ 284, 8 ∤ 824112284.

An integer (q) is divisible by 4 iff q’s last two digits are divisible by 4.

Since 4 ∣ 84, 4 ∣ 824112284.

### Exercise \(\PageIndex{4}\): Divisiblity

1. Let \(a\) be a positive integer such that \(3|(a+1)\) . Show that \(3|(7a+4)\).

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

3. Let \(a\), \(b\), \(c\) and \(d\) be positive integers such that \(4|(abc+d)\), (4|(adc+b)\), \(4|(abd+c)\) and \(4|(bcd+)\). Prove that \(4|(a^2+b^2+c^2+d^2).\)

4. Let \(a\), \(b\), \(c\) and \(d\) be positive integers such that \(ad-bc >1\). Show that at least of these four integers is not divisible by \(ad-bc\).

### Exercise \(\PageIndex{5}\): Middle digit

- 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?
- 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 \(\PageIndex{6}\): True or False

Prove the statements that are true and give counterexamples to disprove those that are false.

Let \(a,b\) and \(c\) be integers.

- If \(a|b\) then \(b|a\).
- If \(a|bc\) then \(a|b\) and \(a|c.\)
- If \(a|b\) and \(a|c\) then \(a|bc\).
- If \(a|b\) and \(a|c\) then \(a|(b+c)\) and \(a|(b-c)\).
- If \(a|(b+c)\) and \(a|(b-c)\) then \(a|b\) and \(a|c\).
- If \(a|b\) and \(a|c\) then \(a|(b+c)\) and \(a|(2b+c)\).
- If \(a|(b+c)\) and \(a|(2b+c)\) then \(a|b\) and \(a|c\)

### Exercise \(\PageIndex{7}\): Induction

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

**Answer**-
use induction to prove the statement.

### Exercise \(\PageIndex{8}\): Induction or by cases

Prove that for all integer \( n\geq 1, \,6\) divides \(n^3-n.\)

**Answer**-
use induction to prove the statement.

### Exercise \(\PageIndex{9}\): True or false

Which of the following statements are true. Prove the statement(s) that are true and give a counterexample for the statement(s) that are false.

If \( a,b,c\) and \(d\) are integers such that \(a < b\) and \(c < d,\) then \(ac < bd \).

If \( a,b,c\) and \(d\) are integers such that \(a < 0 < b\) and \(c < 0 < d,\) then \(ac < bd \).

If \(a,b,c\) and \(d\) are integers such that \(0 < a < b\) and \(c < 0 < d,\) then \(ac < bd \).

If \( a,b,c\) and \(d\) are integers such that \(a < b\) and \(c < d,\) then \(a-d < b-c. \)

**Answer**-
TBD

### Exercise \(\PageIndex{10}\) Tip?

There are 9 waiters and 1 busboy working at a restaurant. The tips are always in whole number of dollars. At the end of the day, the waiters share the tips equally among themselves as far as possible, each getting a whole number of dollars. The busboy gets what is left. How much does he get on a day when the total amount of the tips is $980.

**Answer**-
980 (mod 9) ≡ 8 (mod 9), thus, the busboy would receive $ 8 when the total tip for the day was $ 980.

### Exercise \(\PageIndex{11}\) Day of the week

March 1 on some year is a Friday. On what day of the week will Christmas be that year?

**Answer**-
Christmas falls on December 25th in any given year. The number of days from March 1 till Christmas (430)+(631) -7=299. Note: We consider March 1 as day zero and there are 6 days from December 25 to December 31, thus seven days must be subtracted from the total number of days. Since 299 = 7(42) + 5 and March 1 is a Friday, Christmas will fall five days after Friday, which is a Wednesday. Thus, if in a given year, March 1 is a Friday, Christmas will fall on a Wednesday that year.

### Exercise \(\PageIndex{12}\): Inequality

Solve \(0<199m-335<100\) for all integers \(m\).

**Answer**-
\(m=2\).