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INTEGERS REVIEW EXERCISES Mathematics for Academic Studies 3

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ALFONSO GONZÁLEZ. I.E.S. FERNANDO DE MENA. MATHEMATICS DEPARTMENT

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SECTION 1: Definition of integers, multiple & divisor, and prime number

Definition of integers ():

1. Write from 7 to 23:

2. Represent on a number line the following integers: 5, -4, 2, 0, -1, 1

and put them in order (from least to greatest):

3. Put the following integers in order: -34, 23, 7, 100, -33, 0, 24, -2, 14, -1, 132, -1000

4. Write from -8 to 9:

5. Define the opposite of an integer. Write two examples.

Write the opposite of the following integers as in the example:

a) 7 -(+7)=-7

b) -7

c) 143

d) -25

e) 0

f) -1

g) 45

h) 2

i) -57

6. Define the absolute value of an integer. Give two examples.

Calculate the absolute value of the following integers:

a) |5|=

b) |-3|=

c) |57|=

d) |-23|=

e) |0|=

f) |-1|=

g) |-114|=

h) |12|=

i) |-37|=

j) |5-2|=

k) |1-3|=

l) |-4-3|=

m) 1 3

Multiples & divisors:

7. Write the set of the first six multiples of the following numbers:

a) 6 6,

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b) 15

c) 120

d) 500

8. Write the set of divisors of each number:

a) 12

b) 36

c) 75

d) 23

9. a) Calculate all the divisors of 42 and its positive multiples less than 300.

b) Write all the (positive) divisors of 24 and its two-figure (positive) multiples.

10. a) Write in order all the three-figure (positive) multiples of 75:

75

b) Find all the divisors of 75:

Div(75)




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11. The quotient of two positive integers is 2/3 and their product is 216. What are the numbers? (Advice: factorize

216 and consider each possibility). (Sol: 12 y 18)

Prime and composite numbers:

12. Find the prime factor decomposition of the following integers (and check your solutions):

a) 65 b) 90 c) 125

d) 492 e) 671 f) 1135

g) 1080 h) 441 i) 4950

13. In some cases, we can directly factorize an integer into primes, as in the example:

a) 900=9·100=9 ·10 ·10=32·2 ·5 ·2 ·5=2

2·3

2·5

2

b) 99= (Sol: 32·11)




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c) 1000= (Sol: 23·5

3)

d) 500= (Sol: 22·5

3)

e) 160= (Sol: 25·5)

f) 98= (Sol: 2·72)

g) 250= (Sol: 2·53)

h) 146=

i) 130=

j) 380= (Sol: 22·5·19)

k) 360= (Sol: 23·3

2·5)

14. Define prime number, and give three examples.

Define composite number, and mention three examples.

Justify if the following numbers are prime or composite. In case of being composite, write their decomposition,

as in the following example:

a) 91 As 91 9,.. , it is enough to test possible prime divisors less than 9:

91 is NOT divisible by 2



91 IS divisible by 7 (see the division)

b) 51

c) 79

d) 59

91=7 ·13 is COMPOSITE




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e) 87 (Sol: Composite)

f) 133 (Sol: Composite)

g) 139 (Sol: Prime)

h) 143 (Sol: Composite)

i) 119 (Sol: Composite)

j) 151 (Sol: Prime)

k) 187 (Sol: Composite)

l) 1183 (Sol: Composite)

m) 6513 (Sol: Composite)

n) 127 (Sol: Prime)




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o) 89 (Sol: Prime)

p) 131 (Sol: Prime)

q) 137 (Sol: Prime)

r) 141 (Sol: Composite)

s) 2 135 528

(Sol: Composite)

t) 210

-1 (Sol: Composite)

15. Complete the following sets:

Prime numbers between 45 and 67={

div (44 ) ={

9

between 70 and 130 ={

16. Find all the divisors of 113, explaining your procedure. Is it a prime or composite number? (Sol: Prime)




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17. Construct the sieve of Eratosthenes /ɛrəˈtɒsθəniːz/ for the first 100 natural numbers, showing the

factorization of the composite numbers, as in the examples:

1= Special number

2=

3=

4=

5=

6=

7=

8=

9=

10=

11= PRIMO

12=

13=

14=

15=

16=

17=

18= 2 ·9= 2 ·32

19=

20=

21=

22=

23=

24=

25=

26=

27=

28=

29=

30= 2·3·5

31=

32=

33=

34=

35=

36=

37=

38=

39=

40=

41=

42=

43=

44=

45=

46=

47=

48=

49=

50=

51=

52=

53=

54=

55=

56=

57=

58=

59=

60=

61=

62=

63=

64=

65=

66=

67=

68=

69=

70=

71=

72=

73=

74=

75=

76=

77=

78=

79=

80=

81=

82=

83=

84=

85=

86=

87=

88=

89=

90=

91= 7·13

92=

93=

94=

95=

96=

97=

98=

99=

100=

18. a) Can a number be divisible by 9 but not by 3? Justify the answer and give examples.

PRIME




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b) Is it possible the other way round?

c) Can we conclude that 210 is divisible by 15 without doing the division?




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SECTION 2: LCM (Least Common Multiple) & GCF (Greatest Common Factor) of integers

1. a) Define GCF of two or more integers. Explain the practical rule to calculate it using the prime decomposition.

b) Write the set of divisors of 36 and 24, and mark what is the greatest common factor of both numbers.

c) Factorize both numbers, use the practical rule to obtain the GCF of them, and make sure that you find the

same result.

2. a) Define LCM of two or more integers. Explain the practical rule to calculate it using the prime decomposition.

b) Write the first multiples of 20 and 50, and mark what is the lowest common multiple of both numbers.

c) Factorize both numbers and use the practical rule to obtain the LCM of them (Verify that you find the same

result!).

3. Factorize the following pairs of numbers and find their GCF and LCM. Check, if it is possible, your solution with

the formula LCM(a,b) · GCF (a,b) = a · b , as in the example:

a) 12 and 225

proof: GCF(12,225) ·LCM(12,225)=3 ·900=2700=12 ·225

12=22·3

225=32·5

2

LCM(12,225)=22·3

2·5

2=900

Common and non-common factors to the greatest exponent

12=22·3

225=32·5

2

GCF(12,225)=3

Common factors to the lowest exponent

225 5 45 5 9 3 3 3 1

12 2 6 2 3 3 1




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What is this formula for in practice?

b) 8 and 12 (Sol: LCM=24; GCF=4)

c) 8 and 3 (Sol: LCM=24; GCF=1)

d) 8 and 36 (Sol: LCM=72; GCF=4)

e) 75 and 45 (Sol: LCM=225; GCF=15)

f) 30 and 63 (Sol: LCM=630; GCF=3)

g) 12, 18 and 15 (Sol: LCM=180; GCF=3)




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h) 54 and 36 (Sol: LCM=108; GCF=18)

i) 12, 15 and 40 (Sol: LCM=120; GCF=1)

j) 84 and 120 (Sol: LCM=840; GCF=12)

k) 150 and 225 (Sol: LCM=450; GCF=75)

l) 120, 180 and 300 (Sol: LCM=1800; GCF=60)

m) 24 and 83 (Sol: LCM=1992; GCF=1)

n) 330 and 495 (Sol: LCM=990; GCF=165)




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o) 91 and 245 (Sol: LCM=3185; GCF=7)

p) 56 and 140 (Sol: LCM=280; GCF=28)

q) 91 and 98 (Sol: LCM=1274; GCF=7)

r) 89 and 121 (Sol: LCM=10769; GCF=1)

s) 133 and 49 (Sol: LCM=931; GCF=7)

4. a) Is 113 prime or composite? Explain the procedure that you followed (and write the possible divisions on

the right).

b) Find the GCF and LCM of 113 and 226, that you factored previously. (Sol: Prime; LCM=226; GCF=113)




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SECTION 3: Basic operations with integers

Operations with integers: Additions and subtractions:

1. Simplify, as in the example:

a) – (–14)= 14

b) – (–5)=

c) – (+5)=

d) – (–2)=

e) +(–8)=

f ) +(+6)=

g) – (–43)=

h) +(–1)=

i ) – (+1)=

2. Add or subtract the following integers (It is highly recommended, when it is applicable (/ˈæplɪkəbəl/),

simplifying signs before):

a) 5+15=

b) –5+9=

c) –17+12=

d) –2–15=

e) –23+38=

f) –18–7=

g) –5+20=

h) 40– (–5)=

i ) –2– (–1)=

j ) 12+(–3)=

k) –7– (–5)=

l ) 32+(–6)=

m) |3–7 |=

n) |–1+6 |=

o) |–2–5 |=

3. Do the following additions and subtractions (Simplify signs before, if it is possible):

a) 1+8–7=

b) –5– (–7)+12=–5+7+12= 14

c) 8+13– (–1)=

d) – (–4)–7+(–3)=

e) 12– (–2)–11=

f ) –3+9– (–2)=

g) –5+2– (–3)=

h) 4– (–10)– (–5)=

i) –2+(–1)+14=

j ) 1– (–2)+(–3)=

k) |2–3+6 |=

l ) |–3+2–1 |=

m) 3– |5–2 |=

4. Do the same:

a) 11–8+14–7=

b) –15– (–2)+1–2=

c) 18+3– (–2)–5=

d) – (–14) – (–7)+(–13)+2=

e) 1–2– (–2)–1=

f) –13+19–2+7=

g) –15– (–2)– (–3)+1=

h) 14– (–11)– (–15)–8=

i) –12+(–11)–14+3=

j) 10– (–12)– (–3)– (–5)=




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5. Mental calculation: Do directly the following chained additions and subtractions:

a) 18+6–4+2+1=

b) 12–8–7+5–1=

c) 21+13–8–2+5+6=

d) –23+21–12–5+1–3=

e) –7–4–12–8+4–9+1=

f) 45–20–15+2–7–9+4=

g) –1–2–3–4–5–6–7–8–9=

h) 1–2+3–4+5–6–7+8–9=

6. Calculate the following combined additions and subtractions working out previously the operations inside the

brackets, and always simplifying , as in the example:

a) 11–(8+14–7)=11–15= –4

b) 10–(8–7)+(–9–3)= (Sol: -3)

c) 15–[7–(–3)]= (Sol: 5)

d) (–8–2) –(6–3)= (Sol: -13)

e) –2–[7–(–7)+(–1)]= (Sol: -15)

f) [4–2+(–13)]–(8–3)= (Sol: -16)

g) 12+(15–3)–(–1–18)= (Sol: 43)

h) –4–[34–(–2)]–(–|–5|) = (Sol: -35)

i) 9+3–{–[14–(–5)]–8}= (Sol: 39)

j) 3 2+5 4 1 2 = (Sol: -7)




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Operations with integers: Multiplications and divisions:

7. Multiply:

a) 2 · (–4)=

b) (–3) · (–5)=

c) (–5) ·5=

d) (–1) · (–2)=

e) 6 · (–8)=

f) 3· (–6)=

g) (–7) · (–4)=

h) 3 · (–1)=

i) (–4) ·5 · (–1)=

j) 3 · (–2) ·7=

k) (–4) · (–2) · (–3)=

l) (–1) · (–1)=

m) 3 ·4 · (–6)=

n) 2 · (–2) · (–2)=

o) 3 · (–2) · (–1) ·4=

8. Divide:

a) 15 :5=

b) (–12) :4=

c) (–14) : (–7)=

d) (–42) :6=

e) 21: (–3)=

f) (–18) : (–3)=

g) (–63) :9=

h) 40 : (–5)=

i) (–2) : (–1)=

j) 12 : (–3)=

k) (–75) : (–5)=

l) 32 : (–8)=

9. Do the following combined multiplications and divisions:

a) 2 ·8 :4=

b) (–15) :5 ·12=

c) 8 : ( –2) · (–1)=

d) –(–14) : (–7) · (–3)=

e) 12 : [–(–2)]· (–11)=

f) (–3) ·9 : (–3)=

g) (–75) :5 : (–3)=

h) 64 : (–8) : [–(–4)]=

i) –4 · (–1) :2=

j) 21 : (–7) : (–3)=

10. Mental calculation: Calculate it in your head:

a) 309 :3=

b) 78 :2=

c) 156 :2=

d) 96 :2=

e) 506 :2=

f) 17·3=

g) 5 :2=

h) 78 :3=

i) 147 : (–3)=

j) 306 :6=

k) 102 :3=

l) (–156) :3=

m) 3 ·25=

n) 1012 :2=

o) 78 :3=

p) 84 :21=

q) 36 :3=

r) 102 :3=




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SECTION 4: Combined operations with integers. Hierarchy (/ˈhaɪəˌrɑːkɪ/)

1. Work out step by step the following combined operations with integers:

a) (–3 + 6 + 18) : (–3) = (Sol: -7)

b) (–4) – (–6) : (–3) = (Sol: -6)

c) 5 : (–5) – (–7) · 2 = (Sol: 13)

d) (–11) –3 · (–4) : (–6) – (–9) = (Sol: -4)

e) [2 – (–5) – 3] · (–2) = (Sol: -8)

f) [6 – (–1) – (–13)] : (–5) = (Sol: -4)

g) [(–7 + 5 – 2) – (6 – 8) + 5] : (–3) = (Sol: -1)

h) [(–5) · (–3) · 4 + 12] : [–12 – (–3)] =

(Sol: -8)

i) –4 + 6 · (–2 + 5) : (–9) + 2 · 3 =

(Sol: 0)

j) –18 – [4 + (–6)] : 2 + 5 =

(Sol: -12)

k) {[–4 + 6 · (–2 + 5)] : (–7) + 2} · 3 =

(Sol: 0)

l) 18 : [6 – 3 · (–4 : 2 + 1)] – 3 =

(Sol: -1)

m) (–5) – (–9) – 4 · (–3) : (–2) : (–6) =

(Sol: 5)

n) 3 – 6 : 2 · (–3) : [–2 + (–1)] =

(Sol: 0)




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o) [(–4 + 6 : 3 + 1) · (6 – 4 : 2) + 8] : (–2) =

(Sol: -2)

p) 2 + 4 : 2 – 3 · (–5) + 6 – 3 : (5 – 2 · 3 ) =

(Sol: 28)

q) (–2) · [8 – 6 · (–3 + 12 : 2 ) : (–3) + 1] + (–3) =

(Sol: -33)

r) – 5 2 – 8 – 2 3 – 3 1 1 6 –2 : · :

(Sol: 2)

s) 25 : [– 7 – (– 2)] – (– 5 ) · 4 · |– 2| =

(Sol: 35)

t) 32 8 3 2 81 9 =: · :

(Sol: 7)

u) 14 4 4 12 2 3 1 2 1 · : : :

(Sol: -11)




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v) 2 14 8 2 9 13 9 13 2 : · :

(Sol: 40)

w) 30 15 2 7 12 2 14 1 5 : · : :

(Sol: -5)

x) 20 16 2 7 3 8 4 2 2 3 : · : ·

(Sol: 16)

y) 18 15 33 30 11 13 16 15 5 : : :

(Sol: 1)

z) 218 15 6 2 9 2 3 20 3 121

: : ·

(Sol: -25)

α) 5 3 9 3 9 2 2 8 2 4 : · : ·

(Sol: -20)




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β) 2

3 9 6 2 3 | 4 | 144 15 16 7 · : ·

(Sol: -44)

γ) 4 7 3 36 3 1 2 4 7 4 7 1 2 : · : :

(Sol: -1)

δ) 2 3 02 3 9 13 : 7 9 4 17 1 361 39 39 · ·

(Sol: -683)

ε) 212 12 2 8 2 3 3 81 : : ·

(Sol: 3)



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