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TRƯỜNG ĐẠI HỌC ĐÀ LẠT
F 7 G






GIÁO TRÌNH
LÝ THUYẾT SỐ








VŨ VĂN THÔNG

Z
Z
ϕ(n).
σ(n) τ(n).
¨o µ(n).
Z N
a+x = b
x a + x = b b − a.
Z ·

f : N −→ Z
f
Z f(a) −f(b) a, b ∈ N.
 N × N
(a, b)(c, d) a + d = b + c
N × N/ Z
(m, n)
(m, n).
Z
(a, b)+(c, d)=(a + c, b + d).
(a, b) · (c, d)=(ac + bd, ad + bc).
(Z, +, ·)
0=
(0, 0) , 1=(1, 0).
f : N −→ Z f(n)=
(n, 0).
f
f(a + b)=
(a + b, 0) = (a, 0) + (b, 0) = f(a)+f(b)
f(a · b)=
(ab, 0) = (a, 0) · (b, 0) = f(a) · f(b)
x =
(a, b) ∈ Z.
x =
(a, 0) + (0,b)=(a, 0) −(b, 0) = f(a) −f(b).

n f(n) ∈ Z; N ⊂ Z.
a, b ∈ N,a > b x = (a, b)=(a − b, 0) = f(a − b);
x n = a −b
x = n.

a, b ∈ N,a < b x =
(a, b)=−(b − a, 0) = −f(b − a);
x n = b−a
x = −n.
x =
(n, n) 0.
Z
R x, y ∈ R : x =0,y =0
xy =0.
Z N
N
Z.
Z
Z n
−n Z
N
X g : N −→ X
X
g(n) −g(n),n∈ N. ϕ : Z −→ X, +
n −→ +g(n)

R ≤
x, y ∈ R
∀z ∈ R (x ≤ y ⇒ x + z ≤ y + z)
0 ≤ x, 0 ≤ y ⇒ 0 ≤ xy
Z ≤ x ≤ y y −x ∈ N.
Z N.
Z, ≤

x |x|,

|x| =

x x ≥ 0
−x x ≤ 0
M
M M
M M
M
A = M ∩ N A = ∅ b A
M. −b M
b =min{−x : x ∈ M }.
M 
Z
a b =0, a
b, a
.
: b, c a = bc.
b a, b a, b | a.
a b, b a,
b  a.
6 | 12 ; −5 | 20 ; 7 |−49 ; −8 |−16 ; 15 | 0; 8  12 ;
− 3  8; 4  −9; −12  −18.

a, b
b | a a>0,b>0 1 ≤ b ≤ a.
b | a c | b, c | a.
b | a c =0 bc | ac.
c | a c | b, c | (ma + nb) m, n
a, b b =0.
q, r a = bq + r 0 ≤ r<|b|.

M = {bx : x ∈ Z ; bx ≤ a }
M bq. bq ≤ a
a<bq+ |b|; 0 ≤ r = a −bq < |b|.
a = bq
1
+ r
1
= bq
2
+ r
2
;0 ≤ r
1
,r
2
< |b|.
|b|·|q
1
− q
2
| = |r
1
− r
2
| < |b|; q
1
= q
2
r
1

= r
2
. 
a = bq + r, 0 ≤ r<|b| q r a
b. b | a r =0.
133 21 6 7.
−50 8 −7 6. 50 −8
−6 2. −133 −21 7
14.

1 1
1
1
1
b>1
b>1 n
n = a
k
b
k
+ a
k−1
b
k−1
+ ···+ a
1
b + a
0
k a
j

0 ≤ a
j
≤ b −1
a
k
=0.
n = bq
0
+ a
0
, 0 ≤ a
0
≤ b − 1.
q
0
=0, q
0
b
q
0
= bq
1
+ a
1
, 0 ≤ a
1
≤ b − 1.
q
1
= bq

2
+ a
2
, 0 ≤ a
2
≤ b − 1,
q
2
= bq
3
+ a
3
, 0 ≤ a
3
≤ b − 1,
q
k−2
= bq
k−1
+ a
k−1
, 0 ≤ a
k−1
≤ b − 1,
q
k−1
= b.0+a
k
, 0 ≤ a
k

≤ b − 1.
n = a
k
b
k
+ a
k−1
b
k−1
+ ···+ a
1
b + a
0
0 ≤ a
j
≤ b − 1,a
k
= q
k−1
=0.
n.
n =1 k =0, a
0
=1.
n = a
k
b
k
+ a
k−1

b
k−1
+ ···+ a
1
b + a
0
= c
m
b
m
+ c
m−1
b
m−1
+ ···+ c
1
b + c
0
. (∗)
n b a
0
= c
0
.
a
0
= c
0
n
1

= a
k
b
k−1
+ a
k−1
b
k−2
+ ···+ a
1
= c
m
b
m−1
+ c
m−1
b
m−2
+ ···+ c
1
.
n
1
<n m = k
a
1
= c
1
, ··· ,a
k

= c
k
. 
2.
b =2,
n = a
k
2
k
+ a
k−1
2
k−1
+ ···+ a
1
2+a
0
,
k a
j
0 1,a
k
=0. 
n = a
k
b
k
+ a
k−1
b

k−1
+ ···+ a
1
b + a
0
(a
k
a
k−1
···a
1
a
0
)
b
.
(a
k
a
k−1
···a
1
a
0
)
q
q
b
b
q− b q−

b− b−
610
2
2
106 = 2 ·53 + 0,
53 = 2 · 26 + 1,
26 = 2 · 13 + 0,
13 = 2 · 6+1,
6=2·3+0,
3=2·1+1,
1=2·0+1.
0.
c
0
=0,c
1
=1,c
2
=0,c
3
=1,c
4
=0,c
5
=1,c
6
=1;
1101010.

2003

16
16
2003 = 16 ·125 + 3,
125 = 16 ·7+13,
7=16·0+7.
0.
c
0
=3,c
1
= D, c
2
=7;
7D3.

Z (Z, +, ·)
Z N
Z, ≤
7 −7
a)9 b)99 c) 999 d) 9999 e) 99999
17 −17
a) − 8 b) − 88 c) − 888 d) − 8888 e) − 88888
a, b, c, d a c 0
a | b c | d ac | bd.
a, b, c c =0. a | b
ac | bc.
a, b a | b a
n
| b
n

n.
1955, −1973
ABCDEF
r
r
n
r, n > 1
b<−1
n =0
n = a
k
b
k
+ a
k−1
b
k−1
+ ···+ a
1
b + a
0
k a
j
0 ≤ a
j
≤
|b|−1 a
k
=0.
−7, −17, 61 −2.

n =0
n = a
k
3
k
+ a
k−1
3
k−1
+ ···+ a
1
3+a
0
k a
j
−1, 0, 1
a
k
=0.
13, 40, 121.
n n
a, n a
n
− 1
a =2 n
n
√
n.
p
n

3
√
n n/p 1.
a, b
a b +1 −1.
a
b a b.
a b (a, b).
a
1
,a
2
, ··· ,a
n
a
j
, 1 ≤ j ≤ n.
a
1
,a
2
, ··· ,a
n
(a
1
,a
2
, ··· ,a
n

).
24 84 ±1, ±2, ±3, ±4, ±6, ±12.
(24, 84) = 12. (100, 5) = 5, (0, 44) = 44, (−17, 25) =
1, (17, −289) = 17, (−6, −15) = 3.
(24, −84, 100) = 4, (15, 0, 20, −17) = 1, (10, 20, 30, 40, 55) = 5.

1.
(a, b)=(b, a) (a, b)=(|a|, |b|).
a, b, c (a, b)=d
(a/d, b/d)=1
(a + cb, b)=(a, b)
e e | (a/d) e | (b/d)
k, l a/d = ke b/d = le, a = dek, b = del.
de a b; de ≤ d; e =1.
u a b, u | (a + cb);
u a + cb b. u
a + cb b, u | (a + cb) − cb = a; u
a b. 
a, b ma + nb
a b, m,n
M = ∅
m, n ∈ M m − n ∈ M.
m, n ∈ M,
0=m − m ∈ M, −n =0−n ∈ M, m + n = m −(−n) ∈ M.
a, b ∈ M a b
M. M = {0}
M
M d
M. M d.
m ∈ M m = dk + c, 0 ≤ c<d.

m, dk ∈ M c = m − dk ∈ M. d M
c =0, m d. 
a, b 0 d =(a, b).
M = {ax + by : x, y ∈ Z } d.
M
M
e e M e | a e | b.
e ≤ d. d | (ax + by) x, y ∈ Z d
M, d | e. d ≤ e. 
d =(a, b) a
b.
d a b.
a b d.
x
0
,y
0
∈ Z ax
0
+ by
0
= d. c a
b, c | ax
0
+ by
0
= d.

a
1

,a
2
, ··· ,a
n
,a
n+1
n ≥ 2,
(a
1
,a
2
··· ,a
n
,a
n+1
)=(a
1
,a
2
, ··· ,a
n−1
, (a
n
,a
n+1
)).
c a
1
,a
2

, ···,a
n
,a
n+1
a
n
a
n+1
c (a
n
,a
n+1
). c
a
1
,a
2
, ··· ,a
n−1
, (a
n
,a
n+1
).
c a
1
,a
2
, ··· ,a
n−1

, (a
n
,a
n+1
)
a
1
,a
2
, ··· ,a
n−1
,a
n
,a
n+1
. 
r
0
= a r
1
= b a ≥ b>0.
r
j
= r
j+1
q
j+1
+ r
j+2
, 0 <r

j+2
<r
j+1
j =0, 1, 2, , n −2 r
n+1
=0, (a, b)=r
n
, 0
c = dq + r (c, d)=
(c − qd,d)=(r, d)=(d, r).
r
0
= r
1
q
1
+ r
2
0 <r
2
<r
1
r
1
= r
2
q
2
+ r
3

0 <r
3
<r
2
r
j−2
= r
j−1
q
j−1
+ r
j
0 <r
j
<r
j−1
r
n−2
= r
n−1
q
n−1
+ r
n
0 <r
n
<r
n−1
r
n−1

= r
n
q
n
+0 .
(a, b)=(r
0
,r
1
)=(r
1
,r
2
)=··· =(r
n−2
,r
n−1
)=
(r
n−1
,r
n
)=(r
n
,r
n+1
)=(r
n
, 0) = r
n

. 
(610, −1955).
(610, −1955) = (610, 1955) (610, 1955).
1955 = 610 ·3 + 125
610 = 125 ·4 + 110
125 = 110 ·1+15
110 = 15 ·7+5
15 = 5 · 3+0.
(610, 1955) = 5, (610, −1955) = 5.

(1955, 2003).
2003 = 1955 ·1+48
1955 = 48 ·40 + 35
48 = 35 ·1+13
35 = 13 ·2+9
13 = 9 · 1+4
9=4·2+1
4=1·4+0.
(1955, 2003) = 1, 1955 2003

1
a, b, c (a, b)=1 a | bc
a | c.
(a, b)=1 x, y
ax + by =1. c acx+ bcy = c.
a | bc a | acx + bcy = c. 
p a
1
a
2

···a
k
, a
1
,a
2
, ···,a
k
i, 1 ≤ i ≤ k p | a
i
.
k. k =1
p a
1
a
2
···a
k
a
k+1
p  a
k+1
(p, a
k+1
)=1; p | a
1
a
2
···a
k

. 
n
1
n =2 n+1 > 2
n +1=ab, 1 <a, b<n+1;
a, b
n = p
1
p
2
···p
r
= q
1
q
2
···q
s
p
1
≤ p
2
≤···≤p
r
,q
1
≤ q
2
≤···≤q
s

r = s p
1
= q
1
, ··· ,p
r
= q
s
. 
n>1
n = p
α
1
1
p
α
k
k
, 1 ≤ k, 0 <α
1
, ··· ,α
k
.
p
1
=2<p
2
=3<p
3
=5<p

4
=7<p
5
=11< ···
n =
+∞

k=0
p
α
k
k
,
α
k
≥ 0 0
k.
a =0 b =0, [a, b],
a b.
[a, b]=[b, a] [a, b]=[|a|, |b|].
a
1
,a
2
, , a
k
,
[a
1
,a

2
, , a
k
], a
j
, 1 ≤
j ≤ k.
a, b
a =
+∞

k=0
p
α
k
k
b =
+∞

k=0
p
β
k
k
(a, b)=
+∞

k=0
p
min{α

k
,β
k
}
k
, [a, b]=
+∞

k=0
p
max{α
k
,β
k
}
k
(a, b) · [a, b]=ab.
c =
+∞

k=0
p
γ
k
k
d =
+∞

k=0
p

θ
k
k
k : γ
k
≤ θ
k
.
(a, b)=
+∞

k=0
p
min{α
k
,β
k
}
k
, [a, b]=
+∞

k=0
p
max{α
k
,β
k
}
k

.
(a, b) · [a, b]=
+∞

k=0
p
min{α
k
,β
k
}+max{α
k
,β
k
}
k
=
+∞

k=0
p
α
k
+β
k
k
= ab.

2100 = 2
2

· 3 ·5
2
· 7, 40 = 2
3
· 5
2
2
· 5=20. 2100 40 2
3
· 3 · 5
2
· 7 = 4200.

a
1
x
1
+ a
2
x
2
+ ···+ a
n
x
n
= c
a
1
,a
2

, ··· ,a
n
= c
a, b d =(a, b).
d  c ax + by = c
d | c ax + by = c
x = x
0
+(b/d)m, y = y
0
− (a/d)m, m∈ Z
x = x
0
,y = y
0
x, y ax + by = c.
d | a d | b d | ax + by = c, d  c.
s, t
as + bt = d.
d | c e c = de
c = de =(as + bt)e = a(se)+b(te).
ax + by = c
x = x
0
+(b/d)m, y = y
0
− (a/d)m, m∈ Z
ax + by = c.
ax + by = c ax
0

+ by
0
= c.
a(x − x
0
)+b(y −y
0
)=0,
a(x − x
0
)=b(y
0
− y),
(a/d)(x − x
0
)=(b/d)(y
0
− y).
(a/d, b/d)=1. (a/d) |
(y
0
−y) m y
0
−y =(a/d)m, y = y
0
−(a/d)m.
a(x − x
0
)=b(y
0

− y), x = x
0
+(b/d)m.

15x − 6y =20
(15, −6) = 3  20.
15x −6y = −9 (15, −6) = 3  −9;
x
0
= −1,y
0
= −1 15x −6y =
−9 x = −1 − 2m, y = −1 − 5m, m ∈ Z.

a
1
,a
2
, ···,a
n
n ≥ 2
d =(a
1
,a
2
, ··· ,a
n
).
d  c a
1

x
1
+ a
2
x
2
+ ···+a
n
x
n
= c
d | c a
1
x
1
+a
2
x
2
+···+a
n
x
n
= c
x
1
,x
2
, ··· ,x
n

a
1
x
1
+ a
2
x
2
+ ···+ a
n
x
n
= c.
d | a
j
j, 1 ≤ j ≤ n, d | a
1
x
1
+a
2
x
2
+···+a
n
x
n
= c;
d  c
n =2

a
1
x
1
+ a
2
x
2
+ ···+ a
n
x
n
+ a
n+1
x
n+1
= c.
d =(a
1
,a
2
, ··· ,a
n−1
,a
n
,a
n+1
)=(a
1
,a

2
, ··· ,a
n−1
, (a
n
,a
n+1
)).
d | c
a
1
x
1
+ a
2
x
2
+ ···+ a
n−1
x
n−1
+(a
n
,a
n+1
)y = c
y,
a
n
x

n
+ a
n+1
x
n+1
=(a
n
,a
n+1
)y

a (a, a +1), (a, a +2), (a, a +3).
(a, b)=1 (a + b, a −b) 1 2.
6k −1, 6k +1, 6k +2, 6k +3, 6k +5
k.
(a, b)=1 c | a + b (c, a)=(c, b)=1.
(a, b)=(a, c)=1 (a, bc)=1.
(a
1
,b)=(a
2
,b)=···=(a
n
,b)=1 (a
1
a
2
···a
n
,b)=1.

a
1
,a
2
, , a
n
c (ca
1
,ca
2
, ··· ,ca
n
)=
|c|·(a
1
,a
2
, ··· ,a
n
).
a, b, c, d b>0,d > 0, (a, b)=
(c, d)=1 a/b + c/d b = d.
a, b
(a, b)=








a a = b
2(a/2,b/2) a, b
(a/2,b) a b
(a −b, b) a, b a>b.
(2106, 8318).
a, m, n a>1.
(a
m
− 1,a
n
− 1) = a
(m,n)
− 1.
m, n (f
m
,f
n
)=f
(m,n)
,
f
k
k.
111111, 4849845
p n p
α
| n,
p
α+1

 n, p
α
n, p
α
 n.
p
a
 m p
b
 n p
a+b
 mn.
p
a
 m p
b
 n, a = b p
min{a,b}
 m + n.
a, b, c [a, b] | c
a | c b | c.
p a, n p |
a
n
,n>0, p | a.
a, b (a, b)=1
(a
n
,b
n

)=1 n.
a
n
| b
n
n a | b.
a | c, b | c [a, b] | c.
a
j
| c, 1 ≤ j ≤ k. [a
1
,a
2
, , a
k
] | c.
a, b, c [a, b, a]=
[a, [b, c]], ([a, b],c)=[(a, c), (b, c)], [(a, b),c]=([a, c], [b, c]).
a, b, c
[a, b, c]=
abc(a, b, c)
(a, b)(a, c)(b, c)
.
a, b, c (a, b, c)[ab, ac, bc]=
abc, [a, b, c](ab, ac, bc)=abc.
a, b, c ([a, b], [a, c], [b, c]) =
[(a, b), (a, c), (b, c)].
4k −1,k∈ Z.
12x +15y =50 3x +4y =7
30x +47y = −11 102x + 1001y =1

2x +3y +4z =5; 7x +21y +35z =8; 101x + 102y + 103z =1

x + y + z = 100
x +8y +50z = 156

x + y + z = 100
x +6y +21z = 121
1
x
+
1
y
=
1
14
.