solucionario i

7/25/2019 Solucionario i

1/142

2 Matemtica I

1U N I D A D

1. A = {3; 4; 5; 6; 7; 8}

2. 13 12= 0

23 22= 8 4 = 4

33 32= 27 9 = 18

43 42= 64 16 = 48

53 52= 125 25 = 100

A = {x3 x2/x +}

3. x = {3; 4; 5; 6}

A = {7; 14; 23; 34}

5. 2 A 2 A = F

6. A = {9; 10:11; 12; ...} infinito (V )

B = {6; 7; 8} tiene 3 elementos (F)

C = {2} es unitario (F)

D = { } tiene cero elementos (F)

7. M es unitario

x2 y2= 54; x + y = 18

(x + y)(x y) = 54

18(x y) = 54

x y = 3

8. x + y = 16; x y = 8

(x + y)(x y) = 16 8

x2 y2= 128

9. A = {1; {2}; {1}; 1} = {1; {2}; {1}}

n(P(A)) = 2n(A)= 23

n(P(A)) = 8

10. 2 = 1 x 2

6 = 2 x 3

12 = 3 x 4

20 = 4 x 5

90 = 9 x 10

n(A) = 9

n(P(A)) 1 = 29 1 = 511

11.A = {V; E; R; A; C; I; D}

B = {A; V; T; E; N; I; C; O}

A B = {E; A; C; I}

n(A B} = 4

12.A = {2; 3; 4; 5; 6; 7}

B = {6; 7; 8; 9; 10}

A B = {2; 3; 4; 5; 6; 7; 8; 9; 10}

Suma = 54

4. x + x {1; 2; 3; ... }

1 1

1 + 1=

0

2= 0 5 1

5 + 1= 4

6= 2

3

2 1

2 + 1= 1

3 6 1

6 + 1= 7

5

3 1

3 + 1

=2

4

=1

2

7 1

7 + 1

= 6

8

= 3

4

4 1

4 + 1=

3

5

x 1

x + 1x +

Pg. 11

Clave: a

Clave: c

Clave: d

Clave: b

Clave: e

Clave: b

Clave: b

Clave: b

Clave: c

Clave: c

Clave: c

Clave: b


2/142

Matemtica I

13.A = {4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17}

B = {13; 16; 19; 22}

C = {11; 13; 15; 17}

A B = {13; 16}

A C = {11; 13; 15; 17}

(A B) (A C) = {11; 13; 15; 16; 17}

n[(A B) (A C)] = 5

19.A = {8; 9; 10; 11}

B = {11; 12; 13; 14}

A DB = {8; 9; 10; 12; 13; 14}

A B = {11}

n(A DB) (A B) = 6 1 = 5

20.A = {m; a; n; g; o}

B = {g; o; m; a}

C = {m; a; n; g}

I. A B F

II. B A V

III. {m; a} A V

21.Si A B n(A B) = n(B)

n(B) = 24

4n(A) = 3n(B)

n(A) = 18

18; 24

22.3n 1= 81 = 34n = 5

5m + 1= 625 = 54m = 3

C = {32; 42; 52} = {9; 16; 25}

Suma: 50

23.P Q = {2; 4; 6}

Q R = {1; 7}

Q P = { }

(P Q) (Q R) (Q P) = {1; 2; 4; 6; 7}

24. (A B)' = {6; 5; 7; 8; 9}

(C A') = {2; 5; 7; 8,9}

(A B)' (C A') = {6}

n[(A B)' (C A')] = 1

14.U = {9; 10; 11; 12; 13; 14; 15; 16; 17; 18}

Q = {12; 13; 14; 15; 16}

Q1= {9; 10; 11; 17; 18}

15.5a + 3 = 18 b + 2 = 18

5a = 15 b = 16

a = 3

3b

4a= 3 16

4 3= 4

16.(A B) (B A) = (A B) (A B)

A = {1; 2; 3; 4; 5; 6}

B = {4; 5; 6; 7; 8}

(A B) (A B) = {1; 2; 3; 7; 8}

Suma: 21

17.n(A B) = n(A) + n(B) n(A B)

n(A B) = 20 + 24 4

n(A B) = 40

18.B' = {0; 1; 2; 3; 12}

A = {0; 1; 2; 3; 4; 5; 6; 7}

B' A = {12}

n(B' A) = 1

Clave: d

Clave: c

Clave: d

Clave: c

Clave: c

Clave: d

Clave: c

Clave: e

Clave: b

Clave: b

Clave: c

Clave: b


3/142

4 Matemtica I

1.

x = 20 12

x = 8

a = 22 8

a = 14

Rpta: A) 14B) 8

C) 34

2. n(A) 8 = 30 8 = 22

n(B) 8 = 16 8 = 8

6. 15 3 + 3 + 12 3 + n(A B)' = 28

27 3 + n (A B)' = 28

n(A B)' = 4

7. 95 38 + 38 + n(B A) + 24 = 200

119 + n(B A) = 200

n(B A) = 81

8. 35 10 + 10 + 26 10 + n(A B)' = 80

51 + n(A B)' = 80

n(A B)' = 29

9.

10 + 6 + 8 + x = 28

x = 4

10.

40 + 30 + 75 + x = 210

x = 65

3. T = 42 14 + 14 + 24 14

T = 66 14

T = 52

4. 21 n(A B) + n(A B) + 15 n(A B) = 31

36 n(A B) = 31

n(A B) = 5

n(B A) = 15 5 = 10

5. 48 n(A B) + n(A B) + 58 n(A B) + 5 =

111 n(A B) = 90

n(A B) = 21

n(B A) = 58 21 = 37

Clave: b

Clave: a

Clave: d

Clave: d

Clave: e

Clave: e

Clave: c

Clave: d

Clave: a

Clave: e

Pg. 15

R = 20

12 ax

x

x

S = 22

A = 16

H = 70

10

40

8

75

6

30

L = 14

C = 105

28

210


4/142

Matemtica I

11.

2 x + 90 + x = 240

3x = 150

x = 50

Solo ingls: 2(50) = 100

14.

U = 100 = 40 x + 50 x + 60 x + x

100 = 150 2x

x = 150 100

2x = 25

15.

a = 35 16 = 19

b = 15 13 = 2

c = 18 15 = 3x = 44 35 2 3 3

x = 44 43

x = 1

12.

Luego: 26 + 22 = 48

13.

a = 12 11 = 1

a = 1

b = 18 14

b = 4

c = 15 13

c = 2

a + b + c = 7

Clave: c

Clave: a

Clave: b

Clave: bClave: a

A

A = 35

U

U

60

12

50

9

B

B = 15

C

C = 18

40 x

4a b

60 x

66

50 x

3

c

= 100

= 44

F

I

26

2x

22

x90

27

B

F

100

240

T = 12

a4

6 9

27

b

c

5

P = 187

C = 15

210

40

10


5/142

6 Matemtica I

16.

a = 18 11 a = 7

U = 10 + 2 + 7 + 4 + 3 + 6 + 8

U = 40

18.

a = 26 8 5 4

a = 9

x = 21 9

x = 12

Nos piden b + x + c

b = 31 5 9 12

b = 5

c = 28 4 9 12

c = 3

b + x + c = 5 + 12 + 3 = 20

19.

a = 29 10 m

a + m = 19 ...I

b = 24 13 n

b + n = 11 ...II

c = 17 8 p

c + p = 9 ...III

I + II + III

a + b + c + (m + n + p) = 39

Adems:x + a + b + c + (m + n + p) = 45

Restando:

x = 45 39

x = 6

Clave: b

Clave: c

Clave: bClave: d

F =

A = 26

A = 29= 15

U

U

U

79

21

B = 18

B = 31

M = 2413

10 8

V =

C = 28

G = 17

2

5

m

10

8

a

a

b

b

4

4

p

5

3

a

x

6

x

n

8

c

c

17.a. 12 + 9 + 5 = 26

b. 7 + 4 + 2 = 13

c. 50 12 7 9 6 4 2 5 = 5

20.

8a + a + 3a + 50 = 170

12a = 120

a = 10

8a = 80Clave: b

= 170

8a

M = 9a

3a

50

P

a

U

1. Slo II

Clave: c

Pg.


6/142

Matemtica I

2. a. n(A B) = n(A) n(B) = 25

10.A = {2, 3, 4}

B = {7; 8; 9}

R = {(x, y) A B/y = 2x}

R = {(4; 8)}

4 + 8 = 12

1. 250 + 150 = 400

2. 7 aos

9. A B R = {(1; 3); (5; 7)}

1 3

2 5 7 3 = 21

5 7

93. R = {(2; 2), (5; 5), (6; 6), (9; 9), (12; 12)}

n(R) = 5

4. 1 + 4 + 9 + 16 = 30

5. A = {7}

B = {0; 3; 5}

C = {2; 4; 6; 8}

D = {6; 7; 8}

n(A B) = 3

n(A C) = 4

n(B C) = 12

n(B D) = 9

Clave: a

Clave: d

Clave: b

Clave: c

Clave: e

Clave: d

Clave: c

Clave: b

Pg.

3. a = Joaqun

b = pap

b a = 30 b = 30 + a

a + b = 56

Reemplazamos

a + 30 + a = 56

2a = 56 30

a = 13

dentro de 5 aos: 18

4. Bertha: S/. 72

Patricia: S/. 72 + S/. 45 = S/. 117

Juntos: S/. 72 + S/. 117 = S/. 189

Clave: a

Clave: b

6. R = {(x, y) A B/ x + 3 = y}

7. A = {4; 5; 6; 7; 8}

B = {7; 8; 9; 10; 11; 12; 13; 14}

R = {(x ,y) A B/y = 2x}

R = {(4; 8), (5; 10), (6; 12), (7; 14)}

Ran(R) = {8; 10; 12; 14}

8. A = {1,2,3}

B = {5; 6; 7; 8}

R = {(x ,y) A B/y = x + 3}

R = {(2; 5), (3; 6)}

Dom(R) = {2; 3}

Clave: c

Clave: b

Clave: a

R


7/142

8 Matemtica I

6. M + S + D = 48 ...(I)

M S = D M = S + D ...(II)

(II) en (I):

2M = 48

M = 24

7. 17 20 = 340

19 10 = 190

29 5 = 145

675

11.a + b = 45678 ....(I)

a b = 9856 a = 9856 + b

de (I): 9856 + b + b = 45678

2b = 35822 b = 17911

a = 27 767

12.P + p = 260 ....(I)

P p = 20 P = 20 + p

Luego de (I): 20 + p + p = 260

2p = 240

p = 120

P = 140

13.4 5 a 2 b +

3 c 2 d 7

e 9 0 0 2

a = 7; b = 5; c = 3; d = 7; e = 7

a + b + c + d e = 15

5. R + J = 124

R J = 32

2R = 156

R = 78

J = 46Clave: a

Clave: a

Clave: d

Clave: c

Clave: d

Clave: e

9. 1R = 1 caja 1

A = 3V 5A = 5(3) 15

V = 8B 15V = 15(8) 120

R = 5A 5

141 cajas

8. 1A 24 12 = 288 +

1B 31 8 = 248

536

Clave: d

Clave: e

10. km

68 6 = 408

Clave: c

87+ 408

495

14.abc 2(1000 abc) = 283

abc 2000 + 2abc = 283

3abc = 2283

abc = 761

Clave: a

15.(a + b + c)2= 144

a + b + c = 12

abc

bca

cab

1332

Clave: a


8/142

Matemtica I

Clave: b

Clave: e

Clave: b

Clave: b

Clave: a

Clave: e

Clave: b

Clave: e

Clave: b

Clave: c

Clave: a

Clave: a

16.25 13 = 12

17.224 50 = 1200 tizas

1200 : 16 = 75

18.q = 851

d = 93

93 851 = 79 143

19.5 80 = 400

5 48 = 240

160

20.68 68 = 4624

(68 + 13)(68 + 13) = 6561

6561 4624 = 1937

27.1204 : 43 = 28

28.1176 : 98 = 12 45 + 12 = 57

21. 225= 15

22.48 : 4 = 12

1. a2+ b2= 25

32+ 42= 25

Luego:

33+ 43=

27 + 64 = 91

2. 4 9 = 36

9= 3

26.(a + b) : 2 = 850

a + b = 1700 ...I

4(a b) = 600

a b = 150 ...II

Luego:

a + b = 1700

a b = 150 a = 150 + b

en (I): 150 + b + b = 1700

2b = 1550

b = 775

a = 925

Pg.

Clave: e

Clave: e

23.I = 12

M = 16

Madre = 28 + 19 = 47

Esposo = 47 + 4 = 51

51 16 = 35

24.163 10 = 153

153 : 9 = 17

Clave: c

25.x : 109 = 2 109

x : 109 = 218

x = 218 109

x = 23 762


9/142

10 Matemtica I

Clave: b

Clave: d

Clave: c

Clave: a

Clave: d

Clave: e

Clave: b

Clave: b

8.

4. A = 32 (2 + 3)

A = 32 5 = 27

6.15

7. 327= 31= 327

5. M = 21 5 8 4 : 2 + 32

M = 4 8 2 + 9

M = 32 2 + 9

M = 39

1. Solo II

2.343(5)

342(5)

341(5)

2131(5)

3. [( a . 5) 2] : 4 = 12

[( a . 5) 2] = 48

5 a = 50

a = 10

a = 100

Clave: b

Clave: c

Clave: c

12.30 22021034 33 323

2 3= 2 . 3

2 3= 1

13. [(4)2 2 x 8] +20 360 : 9

0 + 33 : 32= 3

11.7 728521

4 74 . 54

= 74 . 54

7 . 5= 73 . 52

Pg.

21536= 2532

223

= 233

= 24

3

4 3

Clave: a

Clave: d

9. [(10 26 13) 40] 583

[(640 13) 40] 583

627 40 583 = 4

10.2 . 3 6 123213

6 12 123216

= 3

Clave: a

Clave: a

3. 111(3)= 9 + 3 + 1 = 13

111(4) = 16 + 4 + 1 = 21

111(5) = 25 + 5 + 1 = 31

65

4. a = 1 + 2 + 3 = 6

215= 2

3 6 (111)2 3=

12 (111)2 6


10/142

Matemtica I

Clave: a

Clave: d

Clave: b

Clave: c

Clave: e

Clave: a

Clave: a

Clave: c

Clave: c

6. 4345(6)= 4 63

+ 3 62

+ 4 6 + 5 = 864 + 108 + 24 + 5

= 1001

Luego: 1 + 0 + 0 + 1 = 2

7. 3475 8

27 434 8

35 34 54 8

3 2 6 6

S= 6 + 6 + 2 + 3 = 17

8. 8666 7

0 1238 7

6 176 7

1 25 7

4 3

3 + 4 + 1 + 6 + 0 = 14

12.3452(7)= 3 343 + 4 49 + 5 7 + 2 = 1262

1262 9

2 140 9

5 15 9

6 1

1652(9)

15.1106(7)= 343 + 49 + 0 + 6 = 398

398 8

6 49 8

1 6

a = 6

13.si 111(2)= 4 + 2 + 1 = 7 7 5

2 1 12(5)= ab(5)

Luego: a = 1; b = 2

a + b = 3

14.10a + b 10b a = 72

9a 9b = 72

9(a b) = 72

a b = 8

11.5445(6)=

5 63+ 4 62+ 4 6 + 5 = 1253

Luego:

1 2 5 3 = 30

9. 59 = 111011(2)

Suma de sus cifras = 5

Clave: b

5. 3433(5)

Clave: d

10.7849 8

1 981 8

5 122 8

2 15 8

7 1

1 + 7 + 2 + 5 + 1 = 16

Clave: a

16.333(4)= 48 + 12 + 3 = 63

63 5

3 12 5

2 2

m = 2


11/142

12 Matemtica I

Clave: e

Clave: a

Clave: d

Clave: e

Clave: b

Clave: c

Clave: d

Clave: e

Clave: a

Clave: b

24.35(a)= 21(a + 4)

3a + 5 = 2(a + 4) + 1

3a + 5 = 2a + 8 + 1

a = 4

33(4)= 12 + 3 = 15

25.5225(6)= 1080 + 72 + 12 + 5 = 1169

1169 4

1 292 4

0 73 4

1 18 4

2 4 4

0 1

102101

aobaoa

Luego: 2(1+2)

1

= 6

26.3 63+ 5 62+ 5 6 + 3 = 3553(6)

27.500(7)= 49 5 = 245

245 8

5 30 8 6 3

365(8)53= 125

a b

19.a + 2 < 8

a < 6

5 + 4 + 3 + 2 + 1 = 15

20.2 < p < n < m < 6

2 < 3 < 4 < 5 < 6

3 + 4 + 5 = 12

22.2n + 3 + 5n + 4 = n2+ n

n2 6n 7 = 0

n = 7

21.(2n)3n(6)= 3n3(7)

n = 2 323(7)= 147 + 14 + 3 = 164

Luego: 164 6

44 27 6

2 3 4

18.a0(8)= a4(6)

Ba + 0 = 6a + 4

2a = 4

a = 2

Luego: 2(2)2= 8

17.a3(7)= a5(6)

7a + 3 = 6a + 5

a = 2

Luego: aa= 4

Clave: c

23.123(x)= 53(x + 2)

x > 3 x + 2 > 5

x > 3 x > 3

Luego: 123(5)= 25 + 10 + 3 = 38

38 7

3 5 53(7)

x = 5


12/142

Matemtica I

Clave: e

Clave: a

Clave: e

Clave: d

28.2211(6)= 432 + 72 + 6 + 1 = 511

511 8

7 63 8

7 7

x = 8

29.a 82+ a 8 + 5 = (a 3) 122+ 5

64a + 8a + 5 = 144a 432 + 5

432 = 72a

a = 6

31.abc(5) = cba(7)

a 52+ b 5 + c = c 72+ b 7 + a

25a + 5b + c = 49c + 7b + a

24a = 2b + 48c

12a = b + 24c ...(I)

Ademas: a; b; c < 5

a = 2; b = 0; c = 1

a = 4; b = 0; c = 2

Luego: 4 + 0 + 2 = 6

y > x

Adems: 7 > y > x > 4 y = 6

x = 5

565(7) = 5 72+ 6 7 + 5 = 292

292 9

22 32 9

4 5 3

354(9)

1.

2.

3.

Clave: d

Clave: a

Clave: c

II = 180

P = 80

M = 12

IC = 130

G = 60

P = 13

180 x

60

7

130 x

40

8

x

x

x

20

5

210

130

22

Pg.

180 x + x + 130 x = 210

100 = x

60 + 20 + 40 + x = 130

x = 10

7 + 5 + 8 x = 22

x = 2

30.x > 4 y > x >4


13/142

14 Matemtica I

Clave: c

Clave: b

Clave: e

Clave: e

Clave: a

Clave: e

Clave: e

4.

15 a b + a + 2 + b + 12 a c + c + 16 b c = 31

14 = a + b + c

6.

Solo dos tipos de jugo: 47 + 11 + 31 = 89

5.

a + x = 10

b + x = 6 a + b + c + 3 x = 25 ...(I)

c + x = 9

3 + 26 + 28 a x c + c + 27 b x c = 66

18 = a + b + c + 2x ...(II)

De (I) y (II):

a + b + c + 2 x + x = 25

18 + x = 25 x = 7En (I):

a + b + c + 3(7) = 25 a + b + c = 4

Luego:

a + b + c + x = 4 + 7 = 11

L100= 17 JF = 120 JP = 105

JN = 139

1E = 26

3E = 27

L500= 18

L200= 14

2E = 28

a

a

15 a b

26 a b x 28 a x c

27 b x c

12 a c

16 b c

b 47 31

b

c

c

2 18

11

2

31 270

66

3

Pg.

1. Carlos = 80

Fernando = 3(80) = 240

80 + 240 = 320

2. V = 80km/h

D = 80 6 = 480

4. 18 docenas

Precio total de compra = 18 540 = 9720

Ventas:

12,5 = 144 + 6 = 150 150 50 = 7500

18 12,5 = 5,5 = 60 + 6 = 66 66 95 = 3135

2

Venta total = 7500 + 3135 = 10635

Ganancia = 10635 9720 = 915

3. 3 pantalones = 3 70 = 210

2 camisas = 2 45 = 90

Total = 300

Queda: 340 300 = 40


14/142

Matemtica I

Clave: b

Clave: e

Clave: c

Clave: d

Clave: c

Clave: b

Clave: d

Clave: a

Clave: b

Clave: c

5. 2 pares de zapatos = 2 125 = 250

Descuento = 20

Compra = 250 20 = 230

Queda = 300 230 = 70

6. Carlos: 250

Raul: 4(250) = 1000

1000 250 = 750

7. N subconjuntos propios = 255

2n(A) 1 = 255

2n(A)= 256 = 28

n(A) = 8

8. n(A B) = n(A) + n (B) n(A B)

40 = 18 + 26 n(A B)

n(A B) = 4

6. clave tipo A: Docena S/.8

Clave tipo B: Decena S/.6

Compra total:

8 12 + 5 6

96 + 30

126

1. x = 3; 4; 5; 6; 7; 8

A = {16; 25; 36; 49; 64; 81}

16 + 25 + 36 + 49 + 64 + 81 = 271

2. A = {1; 2; 3; {2}} n(A) = 4

N subconjuntos propios = 24 1 = 15

3. 3x + 6 = 15 x = 3

2y 5 = 13 y = 9

x + y = 3 + 9 = 12

4. 3x + 9 = 27 x = 6

y + 12 = 27 y = 15

x2+ y2= 62+ 152= 36 + 225 = 261

5. A = {2; 3; 5} y B = {1; 6; 7}

n(A B) = n(A) n(B)

= 3 3

= 9

Pg. 37

9.

22 + 25 + 18 + x = 168

x = 103

10.

30 + x + 26 + 20 = 100

x = 24

Clave: e

Clave: b

F = 47

AL

B = 43

AR

22

30

18

26

x

20

25

x

168

100


15/142

16 Matemtica I

Clave: b

Clave: c

Clave: b

12.A = ( 41 + 49 + 15) : (11 4) + 1

A = ( 49) : 7 + 1

A = 7 : 7 + 1 = 1 + 1 = 2

13.ab = 2 52+ 3 5 + 4 = 69

62+ 92= 36 + 81 = 117

1.

varones casados = 37

Clave: a

11.

19 + a + 6 + b + 22 + c + 32 + 23 = 145

a + b + c = 43

C

H

A

a19 22

32

b c

6

145

23

Pg. 38

Casados Solteros Total

Varones 37 18 55

Mujeres 12 53 65

Total 49 71 120

Clave: b

Clave: c

Clave: c

2. 213(5)= 2 52+ 1 5 + 3 = 58

58 7

2 8 7

1 1 58 = 112(7)= aab(7)

a = 1; b = 2

2a + b = 2(1) + 2 = 4

3. 1ab4(n)= n31(6)n = 5

1ab4(5)= 531(6)

1 53+ a 52+ b 5 + 4 = 5 62+ 3 6 + 1

125 + 25a + 5b + 4 = 180 + 18 + 1

25a + 5b = 70

5(2) + 4 = 14a = 2; b = 4

a + b = 2 + 4 = 6

4.

total de cajas: 1 + 2 + 2 3 + 2 3 4 = 33

total de libros: 2 + 2 3 + 2 3 4 = 32

total = 33 + 32 = 65

1caja

2rojas

3libros

2

libros

4libros

3amarillos

4azules

Clave: d

5.

a + x + 27 = 52 a + x = 25 ...(I)

x + b + 15 = 55 b + x = 40 ...(II)

52 + b + 15 = 105 b = 38

En (II): 38 + x = 40 x = 2

T = 52

15

N = 55

x12

aF

b15

105


16/142

Matemtica I

2U N I D A D

1. Edad de Rosa : 29 aos

Suma de cifras: 2 + 9 = 11

2. 6300 2

3150 2

1575 3

525 3

175 5

35 5

7 7

1

6300 = 22

. 32

. 52

. 71

Los exponentes son: 2; 2; 2; 1

3. Total: 75 M + V = 75

Mujeres: 58

estudian carrera de ciencias: 8

Varones: 29

estudian carrera de letras: 9

8 = {8; 16; 24; 32; 40; 48 ; 56; 64; 72}

9 = {9; 18; 27 ; 36; 45; 54; 63; 72}Mujeres = 48; Varones = 27

4. D(10) = {1; 2; 5; 10}

1 10 ; 10 1

2 5 ; 5 2

Total: 4 formas

5. 300 9

30 33

3

33 mltiplos de 9

Pg. 46

Clave: d

Clave: b

Clave: b

Clave: e

Clave: a

6. 60 4 60 7 60 28

0 15 4 8 4 2

15 + 8 2 = 21

Clave: d

7. 480 2

240 2

120 2

60 2

30 2

15 3

5 5

1

480 = 25 . 3 . 5

CD(480) = (5 + 1)(1 + 1)(1 + 1) = (6)(2)(2)

= 24

24 = 3 + CDcomp. + 1

20 = CDcomp.

8. 2 + 7 + 5 + 2 + a = 3

a + 16 = 3

2

5

8

amx. = 8

9. m2 = 4

1

3

5

7

9

mmx.= 9

Clave: b

Clave: b

Clave: e


17/142

18 Matemtica I

10.8 + b + 2 + 1 + 3 = 9

b + 14 = 9

4

11.65x482 = 7

231231

+

2 + 24 + 8 x 15 12 = 7

7 x = 7

x = 7

12.1 y + 7 5 + 6 = 11

9 y = 11

y = 9

13.3 + 4 + 6 + n + 7 = 3

n + 20 = 3

1

4

7

1 + 4 + 7 = 12

14.65q = 8

6

Clave: a

Clave: c

Clave: c

Clave: d

Clave: a

15. 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29

= 129

17. CD(N) = (4 + 1)(2 + 1)(a + 1)

75 = 15(a + 1)

5 = a + 1 a = 4

18.90 2

45 3

15 3

5 5

1

90 = 2 . 32 . 5

SD(90) =22 12 1

33 13 1

52 15 1

= 3 . 13 . 6

= 234

19.N = 123= (22 . 3)3 = 26 . 33

CD(N) = (6 + 1)(3 + 1) = 7(4) = 28

28 = 2 + CDComp. + 1

25 = CDcomp.

20.76a = 5 a = 0; 5

a8b = 9 5 + 8 + b = 9

13 + b = 9

b = 5

a + b = 5 + 5 = 10

16. 4 + 6 + 8 + 9 + 10 + 12 + 14 + 15 = 78

Clave: d

Clave: b

Clave: e

Clave: e

Clave: a

Clave: e


18/142

Matemtica I

21.D + C = 237

D: usan lentes: 619

D = 19

C: no usan lentes: 913

C = 13

D = {19; 38; 57; 76; 95; 114; 133 ; 152; ...}C = {13; 26; 39; 52; 65; 78; 91; 104 ; ...}

D = 133 y C = 104

22.D: usan lentes: 619

(133) = 42

D: no usan lentes: 133 42 = 91

C: no usan lentes: 9

13

(104) = 72

C: usan lentes: 104 72 = 32

23.Edad de Maria: 41 aos

Edad de Yolanda: 47 aos

41 + 47 = 88 aos

24.Aprobaron fsica: 15

= 5

Aprobaron Qumica: 17

= 7

MCM(5; 7) = 35

30.N = (3 5)n . 32

N = 3n + 2 . 5n

CD(N) = (n + 2 + 1)(n + 1)

29.11 13 17 = 2431

Clave: d

Clave: b

Clave: c

Clave: b

Clave: e

Clave: e

24 = (n + 3)(n + 1)6 . 4 = (n + 3)(n + 1)

n + 3 = 6 n = 3

3n 2 = 3(3) 2 = 7

25.3n5n6 = 11 + 4

3 n + 5 n + 6 = 11

+ 4 14 2n = 11 + 4

5

n = 5

Clave: c

26.b + 1 + 4 + 3 + 5 + 1 + 6 + a = 9

a + b + 20 = 9

7

a + b = 7 a2+ b2= 52+ 22

a b = 3 = 25 + 4a = 5; b = 2 = 29

27.6a4a38b

38b = 8 b = 4

6a4a38b = 6 + a + 4 + a + 3 + 8 + 4 =

2a + 25 =

a = 1

28.a2b3ba

2b + 15 = 9

b = 6

Clave: d

Clave: e

Clave: c

8

5 a = 0; 5

5 + 2 + b + 3 + b + 5 =

9

9

9

9

Piden: a + b = 5

Piden: a2+ b2= 52+ 62= 61

9

9


19/142

20 Matemtica I

Clave: e

a + 1 = 7 a = 6a + b = 6 + 2 = 8

b + 1 = 3 b = 2

34.N = 304= (2 . 3 . 5)4

N = 24 . 34 . 54

N6= 2 . 3(23 . 33 . 54)

CD(N6) = (3 + 1)(3 + 1)(4 + 1) = 80

35.32m= 25m

CD(32m) = 5m + 1

1 + b + 1 = 5m + 1

b + 1 = 5m m = b + 15

128m= 27m

CD(27m) = 7m + 1

= 7b + 1

5+ 1

=

7 b + 7 + 5

5 =

7b + 12

5

Clave: b

Clave: a36.N = (22 5)n (2 32)

N = 22n5n2 32

N = 22n + 1325n

N3= 3 (22n + 1) 3 5n)

CD(N3) = (2n + 2)(1 + 1)(n + 1)

144 = 2(n + 1)(2)(n + 1)

36 = (n + 1)2

6 = n + 1 n = 5

n

3

n

2

= 5

3

5

2

= 125

25 = 100

37.A = (22 3 5)3 . (32 . 5)2= 26 33 53 34 52

A = 263755

CD(A) = (6 + 1)(7 + 1)(5 + 1) = (7)(8)(6) = 336

B = (235)2 (52)3= 265256= 2658

CD(B) = (6 + 1)(8 + 1) = (7)(9) = 63

Luego:

CD(A) CD(B) = 336 63 = 273

38.N = 41 + 2 + 3 + ... + m= 4

N = (22)

m(m + 1)2 = 2m(m + 1)

CD(N) = m(m + 1) + 1

343 = m2+ m + 1

m = 18

2(18) + 13 = 36 + 13= 49= 7

39.N = 3a . 7a . 11b

CD(N) = (a + 1)(a + 1)(b + 1)3 + 143 + 1 = (a + 1)2(b + 1)

147 = (a + 1)2(b + 1)

49 . 3 = (a + 1)2(b + 1)

72 . 3 = (a + 1)2(b + 1)

Clave: b

Clave: e

Clave: a

31.A = (24)2 . (22 . 5)3= 28 . 26 . 53

A = 214 . 53

B = (22 . 3)2. (3 . 5)4= 24 . 32 . 34 . 54= 24 . 36 . 54

A . B = (214 . 53)(24 . 36 . 54) = 218 . 36 . 57

CD(A . B) = (18 + 1)(6 + 1)(7 + 1) = 1064

1064 = 3 + CDcomp.+ 1

1060 = CDcomp.

32.N = (23 . 3)n . (2 . 32) = 23n . 3n . 2 . 32

N = 23n + 1 . 3n + 2

CD(N) = (3n + 2)(n + 3)

2 + 95 + 1 = (3n + 2)(n + 3)

98 = (3n + 2)(n + 3)

14 . 7 = (3n + 2)(n + 3)

3n + 2 = 14 n = 4

n2 3n + 2 = 42 3(4) + 2 = 6

33.N = (2 . 32)n . (22 . 3) = 2n . 32n . 22 . 3

N = 2n + 2 . 32n + 1

CD(N) = (n + 3)(2n + 2)

67 + 2 + 1 = (n + 3).2(n + 1)

35 = (n + 3).(n + 1) 7.5 = (n + 3)(n + 1)

n + 3 = 7 n = 4

n3 8n + 2 = 43 8(4) + 2 = 64 32 + 2 = 34

Clave: e

Clave: a

Clave: c

m(m + 1)

2


20/142

Matemtica I

40.N = 23 . 5mCD(N) = (3 + 1)(m + 1) = 4(m + 1)

9N = 32 . 23 . 5m

CD(9N) = (2 + 1)(3 + 1)(m + 1) = 12(m + 1)

12(m + 1) 4(m + 1) = 48

8(m + 1) = 48

m = 5

41.M + V = 129

V: juegos matemticos:7

18 18

M: organigramas: 513

13

Resto: software

18 = {18; 36; 54; 72; 90 ; 108; 126}

13 = {13; 26; 39 ; 52; 65; 78; 91; 104; 117}

V = 90 y M = 39

Organigramas:5

13(39) = 15

42.Juegos matemticos:7

18(90) = 35

Organigramas: 15

Software: 129 (35 + 15) = 129 50 = 79

Clave: b

Clave: c

Clave: b

43.N: Nmero de alumnos

420 < N < 600

Hijos nicos: 514

14

Policas escolares: 219

19

N = MCM(14 y 19) = 266k

N = 266(2) = 532

No son hijos nicos: 914

(532) = 342

44.cba = a = 0;5c = c = 2

5b2 = 9 5 + b + 2 = 9 b = 2

Piden: a + b c = 5 + 2 2 = 5

46.A = 17 + 14

B = 17 + 12

A.B = (17 + 14)(17 + 12)

= 17 + 168

= 17 + 17 + 15

= 17 + 15 residuo

47.A = 23 + 16

B = 23 + 19

A.B = (23 + 16)(23 + 19)

= 23 + 304

= 23 + 23 + 5

= 23 + 5 residuo

Clave: c

Clave: d

Clave: d

Clave: a

513

5

45.pq2363p

Clave: a

5

13

13

13

13

Piden: p q = 5 5 = 0

q = 5

4q 20 =

5 + 4q + 6 3 24 9 + 5 =

1 4 3 1 4 3 1

5 q 2 3 6 3 5=

+++ +

p = 0; 5

48.N = 52 7 5n 72n= 52 + n 71 + 2n

CD(N) = (3 + n)(2 + 2n) = (3 + n)2(n + 1)

N35= 5 7(51 + n72n)

2(n + 3)(n + 1) (n + 2)(2n + 1) = 282(n2 + 4n + 3) (2n2+ 5n + 2) = 28

2n2+ 8n + 6 2n2 5n 2 = 28

3n = 24

n = 8

Clave: a


21/142

22 Matemtica I

49.N = 3a 7b

SD(N) = 104

3a + 1 13 1

7b + 1 17 1

= 104

3a + 1 1

2

7b + 1 1

6 = 104(3a + 1 1)(7b + 1 1) = 1248 = 26 48

3a + 1 1 = 26 3a + 1= 27 = 33a = 2

7b + 1 1 = 48 7b + 1= 49 = 72b = 1

N = 3271= 9 7 = 63

6 3 = 18

1. 16 24 2

8 12 2

4 6 2

2 3 2

1 3 3

1 1

MCM = 243 = 48

2. 60 80 2

30 40 2

15 20 5

3 4MCD = 225 = 20

3. 5 6 2

5 3 3

5 1 5

1 1

MCM = 2 3 5 = 30

4. 60 48 2

30 24 2

15 12 3

5 4

5 lapices4 borradores

50.CD(M) = (a + 1)(b + 1) = mn = m(m + 1)

(a + 1)(b + 1) = 11m + 1 ...(I)CD(N) = (2a + 1)(b + 1) = nm = (m + 1)m

(2a + 1)(b + 1) = 11m + 10 ...(II)

(II) (I): (2a + 1)(b + 1) (a + 1)(b + 1) = 9

(b + 1)(a) = 9 = 3 3

b + 1 = 3 b = 2

a = 3

Luego: a b = 3 2 = 1

51.A + B = 26 (I)

MCM (A, B) = 133

Como A y B son nmeros primos, entonces:

MCM (A, B) = A B

AB = 133 (II)

De (II):

A B = 19 17 = 133

A = 19

B = 7

Cumple (I): 19 + 7 = 26

Piden A B = 19 7 = 12

Clave: b

Clave: c

Clave: a

Clave: d

Clave: d

Clave: d

Clave: c

Pg.


22/142

Matemtica I

5. 48 64 2

24 32 2

12 16 2

6 8 2

3 4 2

3 2 2

3 1 3

1 1

MCM = 263 = 192

6. 20 30 2

10 15 2

5 15 35 5 5

1 1

MCM = 223 5 = 60

7. 45 50 2

45 25 3

15 25 35 25 5

1 5 5

1 1

MCM = 2 3252

CD = (1 + 1)(2 + 1)(2 + 1) = 18

8. 280 350 2140 175 5

28 35 7

4 5

MCD = 2 5 7

CD = (1 + 1)(1 + 1)(1 + 1) = 8

Clave: b

Clave: c

Clave: d

Clave: c

9. A = (3 5)3 (225)2

A = 33532452= 243355

B = (2 32)2 (2 3 5)2= 2234223252

B = 243652

MCD (A, B) = 243352= 10 800

10.A = (223)4(2 5)3= 28342353

A = 2113453

B = (233)3 (225)2= 29332452

B = 2133352

MCM (A, B) = 2133453

11.a y b son PESI

a b = 745

MCD(a,b) = 1

745 = MCM(a, b) 1

745 = MCM(a, b)

12.A B = 144 36

A B = 5184

13.36 43 60 2

18 24 30 2

9 12 15 3

3 4 5

MCD = 223 = 12

Reciben 3, 4 y 5

Clave: a

Clave: d

Clave: d

Clave: b

Clave: c


23/142

24 Matemtica I

14.6 8 2

3 4 2

3 2 2

3 1 3

1 1

MCM = 233 = 24

Armando: 24 aos

Ruben: 48 aos

15.1200 1800 3000 2

600 900 1500 2

300 450 750 2

150 225 375 3 50 78 125 5

10 15 25 5

2 3 5

MCD = 233 52= 600

17.45 60 75 3

15 20 25 5

3 4 5

MCD = 3 5 = 15

N de trozos: 3; 4 y 5

18.

1350 + 423 = 1773

Cocientes 3 5 4 2

1350 423 81 18 9

Residuos 81 18 9 0

16.N personas: 2 + 3 + 5 = 10

Clave: c

Clave: a

Clave: d

Clave: c

Clave: d

19.MCM(18m; 24m) = 144

MCM18m

m;

24mm

=144m

72 =144m

m = 2

m3 m2+ 1 = 23 22+ 1 = 5

20.MCD4A2

;6B2

=82

MCD (2A; 3B) = 4

MCD (3 2A; 3 3B) = 3 4

MCD (6A; 9B) = 12

21.A = (23)3(32)2(56) = 29 . 34 . 56

B = (24)2(37)(52)5= 28 . 37 . 510

MCD(A; B) = 28 . 34 . 56= 2m . 3n . 5p

m = 8; n = 4; p = 6

2m 3n + p = 2(8) 3(4) + 6

= 16 12 + 6

= 10

22.E = (2 32)4(23 3)3(2 5)2

F = 243826332252= 21231152

F = (3 5)3 . (2 3)5 . (22 5)4

F = 335325352854= 2133857

MCM(E; F) = 213

311

57

= 2

. 3y

. 5z

= 13; y = 11 ; z = 7

5z y + x = 5(7) 11 + 13

= 35 11 + 13

= 24 + 13 = 37

Clave: e

Clave: d

Clave: a

Clave: a


24/142

Matemtica I

23.abcd = MCM(5; 6; 11) + 4

abcd = 330k + 4

K = 4 abcd = 330(4) + 4

= 1324

24.A = (24 3)2 . (3 5)3= 28 32 33 53

A = 283553

B = (2232)3 (5 7)4 = 26365474

MCD(A; B) = 263553

CD = (6 + 1)(5 + 1)(3 + 1) = 168

168 = 3 + CDcomp+ 1

164 = CDcomp

25.N: N de figuras

N = 5 + 4

N = 8 + 4 N = MCM(5; 8; 13) + 4

N = 13 + 4

5 8 13 2

5 4 13 2 N = 520 + 4

5 2 13 2 N = 520(1) + 45 1 13 5 N = 524

1 1 13 13

1 1 1

26.9 12 15 20 2

9 6 15 10 2

9 3 15 5 3 3 1 5 5 3

1 1 5 5 5

1 1 1 1

MCM = 22325 = 180

Rpta.: 9 de octubre

Clave: b

Clave: c

Clave: d

Clave: d

27.Asisti: xx + y = 275

No asisti: y

x = MCM(6; 8; 11) = 264

y = 275 264 = 11

28.Nmeros consecutivos: x, x + 1; x + 2

x + x + 1 + x + 2 = 69

3x = 66

x = 22

22 23 24 2

22 23 12 2

11 23 6 2

11 23 3 3

11 23 1 11

1 23 1 23

1 1 1

MCM = 233 11 23 = 6072

29.e 10 = 20 3 e = 70d = e 10 d = 70 10 d = 60

c = 80

b + 10 = 80 4 + 60 b = 370

a = b + 10 a = 380

a + b + c d e = 380 + 370 + 80 60 70

= 700

30.MCD(A; B) = 18 A = 18p

B = 18q

(p y q son PESI)

MCM(A; B) = 5814

MCM(18p; 18q) = 5814

18p . q = 5814

Clave: c

Clave: a

Clave: a


25/142

26 Matemtica I

p . q = 323

p . q = 19 17 p = 19

q = 17

A = 18(19) = 342

B = 18(17) = 306

El nmero mayor es 342

Clave: d

2. Origen: 3150 a. C.: 3150

Fin: 31 a. C.: 31

(31) (3150) = +3119

3. Naci 10 a. C.: 10

Muri 54 d. C.: + 54

(+54) (10) = + 64

4. (+29) (6) = + 35

5. (+135) (1158) = +1293

7. (+14) + (56) + (69) + (+91) = 20

8. E = 27

F = +36

E F = (27) (+36) = 63

Clave: b

Clave: d

Clave: a

Clave: c

Clave: c

Clave: c

9. (80) + (+95) + (+17) + (+100) = +132

10.

a + b = 35(a + 3) + (b 8) = a + b 5

= 35 5

= 40

11. (37) (16) = (37) + (+16)

= 21

(21) (42) = (21) + (+42) = +21

12.Sea. x el nmero negativo

x (x) = 48

2x = 48

x = 24

13.Nmeros consecutivos: x; x + 1, x + 2

x + x + 1 + x + 2 = 57

3x + 3 = 57

3x = 60

x = 20

20; 19; 18

Nmero mayor: 18

14. 5 x 4

x = 5, 4, 3; 2; 1; 0; 1; 2; 3; 4

N de valores de x es: 10

Clave: e

Clave: a

Clave: c

Clave: b

Clave: d

Clave: e

Pg. 57


26/142

Matemtica I

15.(475) + (545) + (488) + (262) + (203) + (558)

= +199

16.A = Op(+18) + Op(31)

A = (18) + (+31) = + 13

B = Op(|30|) (33)

B = Op(+30) + (+33)

B = (30) + (+33) = +3

C = Op(39) Op[69 (10)]

C = (+39) Op(59)

C = (+39) (+59) = 20

A + B C = (+13) + (+3) (20) = +36

17.+2580 2,80 + 9,50 1924,30 5,60

+875,90 1076,80 15,60 = +440,30

18. (+156) + (73) + (89) + (+38) + (46) = 14

Perdi 14 puntos

19.Altura: +8848+7354

Desciende: 1494

(+7354) (+3576) = +3778

20. (+3640) + (384) + (124) + (36) + (76)

= +3940

Clave: e

Clave: d

Clave: a

Clave: b

Clave: a

Clave: d

1. Gol a favor: +3

Gol en contra: 2

Puntaje: 17 (+3) + 12(2) (+51) + ( 24) = +27

2. Camisa: +4

Pantaln: 5

15(+4) + 13(5) = (+60) + (65)

= 5

perdi S/. 5

3. A = 253

B = +299

A . B = (253)(+299) = 75647

5. E = (54) (11) = 43

F = (24) : (2) + (27)

F = (+12) + (27) = 15

EF + 2E 3F = (43)(15) + 2(43) 3(15)

= (+645) + (86) + (+45)

= +604

Clave: b

Clave: c

Clave: b

Clave: d

Pg.

4. q = 16

d = 3

D = d . q = (3)(16) = +48

Clave: a


27/142

28 Matemtica I

6. Carla: 9(+10) + 6(3) + 5(1) = +67

Camila: 8(+10) + 5(3) + 7(1) = +58

Santiago: 10(+10) + 4(3) + 6(1) = +82

Benito: 9(+10) + 7(3) + 4(1) = +65

Mayor puntaje: Santiago

7. A . B = P

A . (B 6) = P + 18

A . B 6 . A = A . B + 18

6A = +18

A = 3

2. A = 169

L2 = 169 L = 169

L = 13

2p = 4(13) = 52

1. A= 8 3 + 52 10

A = 8 3 + 25 10

A = 20

3. V = 512

L3= 512

L = 5123

L = 8

4. x3= 1000

x = 10003

x = 10

2x = 2(10) = 20

5. A = (3) (9) (7)

A = (3) + (9) + (+7) = 5

B = (125) (6) + (+81)

B = (125) + (+6) + (+81) = 38

4A 5B = 4(5) 5(38)

= (20) + (190)

= + 170

8. M =

6. (+4)(+36)(+25)(+36) = +129600

7. V F V

8. A = (128) + (+37) 3(3)

A = (128) + (+37) + (9) = 82B = (+102) (+264) + 5(9)

B = (+102) + (264) + (45) = 207

C = (12) : (3) + (90) (17)(2)

C = (+4) + (90) + (34) = 52

A.B 5C = (82)(207) 5(52)

= 16974 + 260 = 17234

9. D + r = 2468 (I)

D = 18 . 136 + r D r = 2448 (II)

De (I) y (II):

D = 2458

Piden: 2 + 4 + 5 + 8 = 19

Clave: d

Clave: e

Clave: d

Clave: c

Clave: a

Clave: e

Clave: c

Clave: e

Clave: d

Clave: d

Clave: d

Clave: c

10.Compr 160 polos a S/. 13

Vende: (160) = 40 a S/. 16

Vende: (160) = 64 a S/. 13

Vende: 160 40 64 = 56 a S/. 10

Venta total = 40(16) + 64(13) + 56(10) = 2032

Venta costo = 2032 160(13)

= 2032 2080

= 48

Perdi S/. 48Clave: a

(225)5 (223)8 (23 5)4

23031259M = = 53 = 125

(23)10 (33)4 (52)3

230312 56

14

25

Pg.


28/142

Matemtica I

10.L1= 19 L2= 16

A1= 192 A2= 16

2

A1= 361 A2= 256

2P1= 4(19) 2P2= 4(16)

2P1= 76 2P2= 64

A1 A2= 361 256 = 105

2P1 2P2= 76 64 = 12

11.E = [(216)1/3+ 10245

491/2]1

E = [ 2163

+ 4 49 ]1

E = [6 + 4 7]1= (9)1= 19

12.

F = 216 1024 5 + 125F = 1120

Clave: e

Clave: d

Clave: e

1. 15 18 24 2

15 9 12 2

15 9 6 2

15 9 3 35 3 1 3

5 1 1 5

1 1 1

360 min = 6h

6:30 am + 6h = 12:30 pm

2. 276 84 2

138 42 2 MCD = 223 = 12

69 21 2

23 7

Longitud del lado: 12 cm

4. 12 15 18 2

6 15 9 2

3 15 9 3

1 5 3 3

1 5 1 5

1 1 1

MCM = 22325 = 180

3. 3 5 7 3

1 5 7 5

1 1 7 7

1 1 1

MCM = 35 7 = 105

Clave: b

Clave: c

Clave: c

Clave: a

Pg. 64

9. A = 3 8 (25 + 1) : [36 (9 2)2]

A = 24 26 : [36 49]

A = 24 26 : (13)

A = 24 + 2 A = 26

Clave: b

5. 420 480 540 2

210 240 270 2

105 102 135 3

35 40 45 5 5

7 8 9

7 + 8 + 9 = 24

Clave: a

6. 2 3 5 2

1 3 5 3

1 1 5 51 1 1

MCM = 2 3 5 = 30

N de muecas = 30k + 1

= 30(2) + 1

= 61Clave: b

Problemas sobre MCD y MCM


29/142

30 Matemtica I

3. Problemas bien resueltos: x (+4)

problemas mal resueltos: 50 x (1)

x(+4) + (50)(1) (x)(1) = 150

4x 50 + x = 150

5x = 200

x = 40

mal resueltos: 50 40 = 10

Clave: c

2.1a2b3ba4b...9 a10b = 72

8

9

10b = 8 b = 41 + 2 + 3 + ... + 9 + 1 + a + a + ... + a +

5 sumandos

b + b... + b = 9

5 sumandos

1 + 92

9 + 1 + 5a + 5b = 9

46 + 5a + 5(4) = 9

5a = 9 66

5a = 9 ( 9 + 3)

5a = 9

3

18

a = 3

a b = 3 4 = 12Clave: b

1. Lunes: +15

martes: 12

miercoles: 19

jueves: +10

viernes: +16

sabado: 18

domingo: + 9

total: + 1

gan S/. 1

2. Temperatura actual: 6c

temperatura inicial: 6 13c = 7c

Clave: d

Clave: e

3. Stano 3: 3

3 + 9 = +6

piso 6

4. +594 + (647) = 53

5. Preguntas correctas: x (+10)

preguntas incorrectas: 20 x (1)

x(+10) + (20 x)(1) = 112

10x + 20(1) (x)(1) = 112

11x = 132

x = 12

piden 20 x = 20 12 = 8

6. +580 + 295 476 + 142 = +541

se encuentra a una altura de 541 m

1. N = 240 15n

n = 243n + 15n + 1

CD(N) = (4 + 1)(n + 2)(n + 2)3 + 176 + 1 = 5(n + 2)2

180 = 5(n + 2)2

36 = (n + 2)2

n + 2 = 6 n = 4

Clave: b

Clave: d

Clave: a

Clave: c

Clave: d

Pruebas nacionales e internacionalePg. 65 Pg.

Problemas con nmeros enteros


30/142

Matemtica I

4. a + 1 + a 1 + a + b = 3

3a + b = 3

b = 3

para que tenga un nmero impar de divisores, su D.C.:

(a + 1)(a 1)ab = p1 p2 ... pn(a + 1)(a 1)ab = N2

1110a + 900 + b = N2

N2= b + 3 = 3 + 3

N2= 9

Clave: c

5. Sudamericanas: 310

no sudamericanas: 520 310

= 210

n peruanos = 2502

= 105

6. Notemos que:

2m n = 3m (m + n) = 3m 5 = 3(m 2) + 1

como m puede tomar cualquier valor entero , en-tonces 2m n puede llegar a ser cualquier mltipode 3 ms 1. Luego entre las alternativas, vemos quelos nmeros 5, 1, 2 y 7 son mltiplos de 3 ms1, mientras que 2 no es mltiplo de 3 ms 1.

Rpta.: 2

1.

Edad de Alfredo: 41suma de cifras: 4 + 1 = 5

2. 6 8 2

3 4 2

3 2 2

3 1 3

1 1

MCM(6; 8) = 23 3 = 24

3. 75 100 5

15 20 5

3 4

MCD(75; 100) = 52= 25

Clave: b

Clave: c

Clave: c

Clave: a

Clave: c

b = 3 b = 6 ; 9

a = 3 a = 3 ; 6 ; 9

Luego: a = 6 ; b = 9

Piden: a + b = 6 + 9 = 15

2K1 2K2 2Kn

4. 292 232= 841 529

= 312

5. 180 17 450 17

10 10 110 26

8

26 10 = 16

6. A = (253) + (+237) + (+86) + (38)

A = +32

B = (114) + (+216) + (154) + (+102)

B = +50

B A = (+50) (+32) = +18

Clave: e

Clave: c

Clave: b

7. A = 162

B = +126

A + B = (162) + (+126) = 36

Clave: e

Pg. 68


31/142

32 Matemtica I

8. A = 289

L2= 289

L = 289L = 17

2p = 4(17) = 68

9. Aprobaron historia:16

6

aprobaron geografia:1

1111

total de alumnos: MCM(6,11) = 66

Clave: b

Clave: b

10.390 420 510 2

195 210 255 3

65 70 85 5 5

13 14 17

Total de trozos: 13 + 14 + 17 = 44

11.A . B = 112

4A3B = 12AB = 12112

= 1344

12.N: N de monedas antiguas

500 < N < 600

N = 6 + 5

N = 9 + 5

N = 11 + 5

N = MCM6;9;11 + 5N = 198 + 5

N = 1983 + 5 = 599

1. p2+ (p + 1)2+ 80 = abc = q2

p y q deben tener 2 cifras

p = {11; 13; 17; 19; ...}

112+ 122+ 80 = q2

q = {19, 23, 29, 31}

37 tiene 4 cifras

p2+ (p + 1)2= 192 80 = 281 ( )

p2 (p + 1)2= 232 80 = 449 ( )

p2+ (p + 1)2= 292 80 = 761 ( )

p = 19 q = 29q = 31 no cumple

suma de cifras de p: 1 + 9 = 10

2. MCD (A, B) = 24

3. MCD (2520; 2000) = 40

Clave: a

Clave: e

Clave: a

Clave: b

Clave: a

Clave: c

Adems:

200 < 24k < 300

8,3 < K < 12,5

k = 9 ; 10 ; 11 ; 12

(9; 10), (9; 11), (10; 11), (11; 12)

Hay 4 parejas de nmeros.

N total de murales es:

115 3 = 345

Luego, la mnima cantidad de trabajadores es:

+ 1 + 1+ = 115

20024

30024< k

5

8 (a < 24)

a > 15

a = (16; 17; 18; 19; 20; 21; 22; 23)

"a" puede tomar 8 valores.

14.MCM(3, 6, 8, 12) = 24

4

3=

32

24;

7

6=

28

24;

5

8=

15

24;

11

12=

22

24

4

3;

7

6;

11

12;

5

8

El menor es5

8 .

15.32 18

18 1 114

1814

16.Pantalones: 12

Camisas: 15

Chompas: 9

Total: 36

Pantalones

Total=

12

36=

1

3

17.total = 4 + 6 + 7 = 17

Total

Jorge=

17

6

18.Un da: 24h

Transcurrido:3

4(24 h) = 18h

22h 18h = 4h

Clave: a

Clave: d

Clave: c

Clave: e

Clave: a

Clave: c

19.2580

12600=

129

630=

43

210

Luego: 43 + 210 = 253

20.7k

8k7k + 8k = 135

15k = 135

k = 9

Numerador: 7(9) = 63

21. 145

180=

29

36=

a

b

a = 29

b = 36

a + b = 29 + 36 = 65

22.Nias: 24

Nias: 32

Total : 56

NiosTotal

= 2456

= 37

23.5k

6k

(5k)(6k) = 270

30k2= 270

k2= 9k = 3

5(3)6(3) = 1518

Denominador: 18

24.1

6 14

N

112

solucionario i

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