## Math 6 Chapter 2 Lesson 13: Multiples and divisors of an integer

## 1. Summary of theory Tóm

### 1.1. Multiples and divisors of an integer

– For \(a, b \in Z\) and \(b \ne 0\) . If there is an integer q such that a = bq, then we say that a is divisible by b. We also say that a is a multiple of b and b is a divisor of a.

**Example 1:** -9 is a multiple of 3 because (-9) = 3.(-3)

**– Attention:**

- If a = bq (b ≠ 0) then we also say a divided by b gives q and write a:b = q.
- The number 0 is a multiple of every non-zero integer.
- The number 0 is not a divisor of any integer.
- The numbers 1 and -1 are the divisors of all integers.
- If c is both a divisor of a and a divisor of b, then c is also called a common divisor of a and b.

**Example 2:**

The divisors of 8 are: -8; -4; -2; -first; first; 2; 4; 8.

The multiples of 3 are: 0; 3; 6; 9; -3; -6; -9;…

### 1.2. Nature

- If a is divisible by b and b is divisible by c, then a is also divisible by c.

\(\left. \begin{array}{l}a \vdots b\\b \vdots c\end{array} \right\} \Rightarrow a \vdots c\)

- If a is divisible by b, then the multiple of a is also divisible by b

\(\left. \begin{array}{l}a \vdots b\\m \in \mathbb{Z}\end{array} \right\} \Rightarrow am \vdots b\)

- If two numbers a and b are divisible by c, their sum and difference are also divisible by c

\(\left. \begin{array}{l}a \vdots m\\b \vdots m\end{array} \right\} \Rightarrow (a + b) \vdots m,\,\,(a – b) \vdots m\)

## 2. Illustrated exercise

**Question 1: **a) Find four multiples of -3; 3

b) Find multiples of -15, knowing that they are between 100 and 200.

__Solution guide__

a) Multiples of -3 and 3 are of the form 3k where \(k \in \mathbb{Z}\)

There are four multiples of -3; 3 is -6, 6, -12, 12.

b) Between 100 and 200 multiples of -15 are the following numbers 105, 120, 135, 150, 165, 180, 195.

**Verse 2:** Let the set A ={7; 8; 9; 10} and B = {4; 5; 6}.

a) How many sums of the form a + b can be made with \(a \in A,b \in B.\)

b) How many of the above sums are divisible by 2?

c) Write a set of elements of the form ab with \(a \in A,b \in B\) in the above set, how many elements are multiples of 5.

__Solution guide__

a) C = {a + b| \(a \in A,b \in B\)}

C = {11, 12, 13, 14, 15, 16}

Up to 6 totals can be made.

b) There are three numbers that are divisible by 2 which are 12, 14, 16

c) T = {28, 35, 42, 32, 40, 48, 36, 45, 54, 50, 60}

In the set T whose elements are multiples of 5 are: 35, 40, 45, 50, 60.

**Question 3: **Prove that \(S = 2 + {2^2} + {2^3} + {2^4} + {2^5} + {2^6} + {2^7} + {2^8} + {2^9}\) is a multiple of (-41).

__Solution guide__

\(S = (2 + {2^2} + {2^3}) + {2^3}(2 + {2^2} + {2^3}) + {2^6}(2 + { 2^2} + {2^3})\)

\(S = 41(2 + {2^2} + {2^3}) \Rightarrow S \vdots ( – 41)\)

So S is a multiple of -41

## 3. Practice

### 3.1. Essay exercises

**Question 1**: Find multiples of -13 greater than -40 and less than 40

**Verse 2: **Find all divisors of -15 and 54

**Question 3: **Find \(a \in \mathbb{Z}\) such that

a) 2a – 7 is divisible by a – 1

b) a + 2 is the divisor of \({a^2} + 2\)

**Question 4: **Find \(a,b \in \mathbb{Z}\) such that (a – 3) b – a = 5.

**Question 5:** Let a and b be two integers other than 0. Prove that: If a is a multiple of b and b is a multiple of a, then a = b or a = -b.

### 3.2. Multiple choice exercises Bài

**Question 1: **Let a, b ∈ Z and b ≠ 0. If there is an integer q such that a = bq then:

A. a is the divisor of b

B. b is the divisor of a

C. a is a multiple of b

D. Both B and C are correct

**Verse 2: **The multiples of 6 are:

A. -6; 6; 0; 23; -23

B. 132; -132; 16

C. -1; first; 6; -6

D. 0; 6; -6; twelfth; -twelfth; …

**Question 3: **The set of divisors of -8 is:

A. A = {1; -first; 2; -2; 4; -4; 8; -8} B. A = {0; ±1; ±2; ±4; ±8}

C. A = {1; 2; 4; 8} D. A = {0; first; 2; 4; 8}

**Question 4:** How many divisors of -24 . are there?

A. 9

B. 17

C. 8

D. 16

**Question 5: **The set of all multiples of 7 whose absolute value is less than 50 is:

A. {0; ±7; ±14; ±21; ±28; ±35; ±42; ±49}

B. {±7; ±14; ±21; ±28; ±35; ±42; ±49}

C. {0; 7; 14; 21;28; 35; 42; 49}

D. {0; 7; 14; 21; 28; 35; 42; 49; -7; -14; -21; -28; -35; -42; -49; -56; …}

## 4. Conclusion

Through this lesson, you should know the following:

- The concept of multiples and divisors of an integer.
- Do related exercises.

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