solucionario i
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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
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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.
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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
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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
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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
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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
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