26
66*.
Agar
p
> 2 tub son bo’lsa,
4
р
ning qiymati
4
1
−
p
yoki
4
3
−
p
ga tengli-
gini isbotlang.
67*.
a
sonni
m
ga bo’lganda qoldiq
r
bo’lsa,
m
r
a
m
a
−
=
tenglikni isbotlang.
68*.
Agar
m
toq son bo’lsa,
2
1
2
−
=
m
m
ni isbotlang.
69*
. Tenglamani yeching:
[ ]
[
]
[ ]
[ ]
.
)
;
4
3
)
;
1
3
)
;
2
)
2
2
2
x
x
d
x
x
c
x
x
x
b
x
a
=
=
+
=
−
=
70*.
10
6
va 10
7
sonlar orasida 786 ga karrali bo’lgan nechta natural son bor?
71*.
1000 kichik natural sonlardan nechtasi 5 va 7 ga bo’linadi?
72*.
100 dan katta bo’lmagan natural sonlardan nechtasi 36 bilan o’aro tub?
73.
1000! ning kanonik yoyilmasida 11 nechanchi darajada keladi?
74.
1964! soni nechta nol bilan tugaydi?
75.
2311 dan oshmaydiganva 5, 7, 13, 17 larga bo’linmaydigan butun musbat
sonlar soni nechta?
76.
Nayti
kolichestvo sel
ы
x polojiteln
ы
x chisel, ne prevosxodya
щ
ix 110 i
vzaimno prost
ы
x s chislom 36.
77.
12317 dan katta bo’lmagan va 1575 bilan o’zaro tub bo’lgan butun mus-
bat sonlar sonini toping.
78.
1000 dan katta bo’lmagan va 363 bilan o’zaro tub bo’lgan butun musbat
sonlar sonini toping.
79.
r
n
! ning kanonik yoyilmasiga
p
tub son nechanchi darajada keladi?
80.
Sonlarni kanonik yoyilmasini toping:
a)
10! ;
b
) 15! ;
c)
20! ;
d
) 25! ;
e
) 30! .
81.
!
10
!
10
!
20
ni kanonik yoyilmaini toping.
82*.
α
ning shunday eng katta qiymatini topingki, bunda
−
⋅
=
α
7
1000
...
102
101
N
butun son bo’lsin.
83*.
(2
m
+1)!!
ning kanonik yoyilmasida
p
tub son nechanchi darajada
bo’lishini aniqlang.
84*.
( )
,
0
,
x
f
y
b
x
a
≤
≤
≤
≤
egri chiziqli trapesiyada butun koordinatali nu-
qtalar soni nechta? Bu yerda
a
va
b
– natural sonlar;
f (x)
– berilgan kesmada uzluk-
siz va nomanfiy funksiya.
85.
x
2
+
y
2
= 6,5
2
doirada nechta butun koordinatali nuqta bor?
86*.
Agar (
a
, 4) = 1 bo’lsa,
(
)
2
1
3
4
3
4
2
4
−
=
+
+
a
a
a
a
tenglik to’g’riligini isbtolang.
27
87*.
Agar (
a, m
) = 1,
m
≥
2,
a
≥
2 bo’lsa,
(
)(
)
2
1
1
)
1
(
...
2
−
−
=
−
+
+
+
a
m
m
a
m
m
a
m
a
tenglik to’g’riligini isbotlang.
88*
.
x
ning qanday qiymatlarida
[ ]
1
2
2
=
−
x
x
tenglik o’rinli.
89*
.
−
=
1
m
x
m
x
tenglamani yeching, bu yerda
m
= 2, 3, 4...
90
. Qanday shartlar bajarilganda [
ax
2
+
bx
+
c
] =
d
tenglama yechimga ega
bo’ladi, bu yerda
a
≠
0,
d
∈
Z
.
91
. Toping:
{ }
{ }
−
2
1
2
)
;
7
)
;
3
8
)
;
6
,
2
)
d
c
b
a
.
92
. Berilgan sonlarni natural bo’luvchilari va ular yig’indisini toping:
a
) 375 ;
b
) 720 ;
c
) 957 ;
d
) 988 ;
e
) 990 ;
f
) 1200.
93
. Berilgan sonlarning barcha bo’luvchilarini toping:
a
) 360 ;
b
) 375.
94*
.
S
(
m
) = 2
m
– 1 sharti qanoatlantiruvchi natural
m
sonlar cheksiz ko’pligini
isbotlang.
95*
. Agar (
m, n
) > 1 bo’lsa,
τ
(
m n
) yoki
τ
(
m
)
τ
(
n
) lardan qaysisi katta, S(
mn
)
va S(
m
) S (
n
) larchi?
96
. Agar
m
= 1968 bo’lsa,
τ
(
m
), S (
m
),
δ
(
m
) larni toping.
97*
. O’zining natural bo’luvchilari ko’paytmasiga teng bo’lgan barcha natural
sonlar to’plami barcha tub sonlar to’plami bilan ustma-ust tushishini isbotlang.
98*
.
a
natural sonning barcha natural bo’luvchilarining
n
-darajasi (
n
∈
Z
)
yig’indisi
S
n
(
a
) formulasini keltirib chiqaring.
99
. Toping:
a
)
S
2
(12);
b
)
S
2
(18);
c
)
S
2
(16).
100
. 28, 496, 8128
sonlar mukammal, ya’ni o’zining bo’luvchilari
yig’indisining yarmiga tengligini isbotlang.
101*
.
Yevklid teoremasi
ni isbotlang: 2
α
(2
α
+1
- 1) ko’rinishdagi juft natural son-
lar mukammal sonlardir, bu yerda 2
α
+1
– 1 – tub son.
102*
.
Eyler teoremasi
ni
isbotlang: 2
α
(2
α
+1
- 1) ko’rinishdagi natural sonlar,
yagona mukammal juft sonlardir, bu yerda 2
α
+1
– 1 – tub son.
103*
.
Ferma masalasi
: 2
α
⋅
r
1
r
2
ko’rinishdagi shunday eng kichik son top-
ingki, uning barcha bo’luvchilari yig’indisi o’zidan uch marta katta bo’lsin, bu yerda
r
1
va
r
2
– tub sonlar.
104*
. Shunday son topingki, uning ikkita tub bo’luvchisi bo’lib,
barcha
bo’luvchilarning soni 6 ta yig’indisi 28 ga teng bo’lsin.
105*
. Natural son ikkita tub bo’luvchiga ega. Shu son kvadratining barcha
bo’luvchilari soni 15 ta bo’lsa, uning kubi nechta bo’luvchiga ega?
28
106*
. Natural son ikkita tub bo’luvchiga ega. Shu son kvadratining barcha
bo’luvchilari soni 81 ta bo’lsa, uning kubi nechta bo’luvchiga ega?
107*
. Isbotlang:
n
n
n
n
d
d
d
d
d
d
d
d
N
1
1
...
1
1
...
1
2
1
1
2
1
+
+
+
+
+
+
+
+
=
−
−
,
bu yerda
d
1
, d
2
,…, d
n
–
N
sonning barcha bo’luvchilari.
108.
* Agar
N
=
a
α
b
β
…
m
µ
(a, b,., m
∈
Z
)
bo’lsa,
shu sonni ikkita son
ko’paytmasi shaklida necha xilda yozish mumkin?
109.
*
N
= 2
α
5
β
7
ϒ
son berilgan
.
Agar
5
N
N
dan kichik 8 ta bo’luvchiga, 8
N
–
N
dan katta .
110.
N
= 2
x
3
-y
5
z
son berilgan. Agar
N
ni 2 ga bo’lsak, yangi sonning
bo’luvchilari
N
ning bo’luvchilaridan 30 ta kam; agar
N
ni 3 ga bo’lsak, yangi son-
ning bo’luvchilari
N
ning bo’luvchilaridan 35 ta kam; agar
N
ni 5 ga bo’lsak, yangi
sonning bo’luvchilaridan 42 ta kam. Shu sonni toping.
111.
Agar biror son to’la kvadrat bo’lishi uchun faqat va faqat uning
bo’luvchilari soni toq bo’lishini isbotlang.
112.
Quyidagilarni aniq qiymatini hisoblang:
a)
π
(4);
b
)
π
(7);
c
)
π
(10);
d
)
π
(12);
e
)
π
(25);
f)
π
(50);
g
)
π
(200);
h
)
π
(500).
113.
( )
nx
x
x
l
≈
π
formula yordamida quyidagilarni taqribiy qiymatini va na-
tijaning nisbiy xatosini toping:
a
)
π
(50), b)
π
(100);
c
)
π
(500).
114*.
Cheb
ы
shev tengsizligi yordamida
( )
(
)
+∞
→
→
x
x
x
0
π
ni isbotlang.
115*.
Ixtiyoriy
p tub son uchun
(
)
( )
р
р
р
р
π
π
<
−
−
1
1
o’rinli, lekin
m
- murakkab
son bo’lsa,
( )
(
)
1
1
−
−
<
m
m
m
m
π
π
o’rinligini ko’rsating.
116.
Toping:
( )
( )
( )
(
)
(
)
(
)
4320
)
;
1500
)
;
1200
)
;
988
)
;
720
)
;
375
)
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
f
e
d
c
b
a
117.
Ko’paytma qiymatini topmasdan ko’paytuvchilarning
Eyler funksiyasini
qiymatini toping:
a
)
ϕ
(5
⋅
7
⋅
13) ;
b
)
ϕ
(12
⋅
17);
c
)
ϕ
(11
⋅
14
⋅
15
⋅
);
d
)
ϕ
(990
⋅
1890).