number-theory

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F
ก
““““ F F”””” F 4
F
F F F F . . 2537
www.thai-mathpaper.net

F F ˈ F 4 15 F F F
ก F ก F F ก ˈ F กF กF ก F
ก ก ก ก ก ก ก ก ก F
ก ก Fก F F กF ก F ก F F
F F F F ก F F F F
F ก 1 ก ก ก ก F ก F F F
2 ก F F ก ก ʽ F 3 F ก
F F
ˈ F F F ˈ Fก F F F F กก F ก F F F
F F F ก F ก F F F F F F
F
22 . . 2549

ก ก
ก F F ˈ ก F F ก
F ก ก ก ก (congruence) F F F F ก F
ก ก ก ก F ˈ 4 F ก ก F F
F F F ก F ˈ
F F F F ก ก F ก F F F F F F
F
30 . . 2549

1 ก 1 8
1.1 ก 1
1.2 F ก 4
1.3 F F 7
2 9 13
2.1 9
2.2 ก ก 11
3 ก F F 15 21
3.1 ก F F 15
4 ก ก ก ก F 23 37
4.1 ก 23
4.2 ก ก 24
4.3 ก ก 28
4.4 ก ก F 32
ก 37

1
ก
1.1 ก
ก 1.1 F a b F ก F F a | b ก a b F F ก F
F a F b
F 1.1 3 | 12 ก 4 F 12 = (3)(4)
3 | 12 ก 4 F 12 = ( 3)( 4)
( 3) | ( 12) ก 4 F 12 = ( 3)(4)
3 | ( 12) ก 4 F 12 = (3)( 4)
F ก F a, b, c ˈ a, b ≠ 0
F a | b b | c
F F b = am c = bk m, k ˈ
c = (am)k = a(mk) mk ˈ
a|c
F ก F a, b, c ˈ F ˈ F
F a | b a | c
b = ap c = aq p, q ˈ
1.1
ก F a, b ˈ a ≠ 0 ก F F a b ก F b = ac
c
1.1
ก F a, b, c ˈ a, b ≠ 0 F a | b b | c F a | c
1.2
ก F a, b, c ˈ F ˈ F F a | b a | c F a | (bx + cy)
x, y

2 F
F F bx = (ap)x = a(px) -----(1)
cy = (aq)y = a(qy) -----(2)
x, y ˈ
(1) + (2); bx + cy = a(px) + a(qy) = a(px + qy)
px, qy ˈ
1.1 F F a | (bx + cy) F ก
F bx + cy ก F ก F (linear combination) x, y
F ก ก F
F ก F a, b, c ˈ F ˈ F
F a | b F F k F b = ak
F F bc = (ak)c = a(kc) kc ˈ
1.1 F F a | bc F ก
F 1.2 ก n F 7 | (23n
1)
ก F F ก F ก F ก ก F ก
F
: F n = 1 F F 7 | (23(1)
1) = 7 | 7 ˈ
P(1) ˈ
: F n = k F P(k) ˈ
F 7 | (23k
1) F P(k + 1) 7 | (23(k + 1)
1)
ก 23(k + 1)
1 = 23k
⋅ 23
1
= 8 ⋅ 23k
1
= (7 + 1) ⋅ 23k
1
= 7 ⋅ 23k
+ 1 ⋅ 23k
1
= 7 ⋅ 23k
+ (23k
1)
ก F F 7 | (23k
1) F F 7 | 7 ⋅ 23k
P(k + 1) ˈ
ก F F F 7 | (23n
1) ก n
1.3
F a, b, c ˈ F ˈ F F a | b F a | bc

F F 3
F 1.3 ก n F 24 | (2 ⋅ 7n
+ 3 ⋅ 5n
5)
F ก F
: F n = 1 F F 24 | (2 ⋅ 71
+ 3 ⋅ 51
5) = 24 | 24
P(1) ˈ
: F n = k F 24 | (2 ⋅ 7k
+ 3 ⋅ 5k
5)
F 24 | (2 ⋅ 7k + 1
+ 3 ⋅ 5k + 1
5)
2 ⋅ 7k + 1
+ 3 ⋅ 5k + 1
5 = 2 ⋅ 7k
⋅ 7 + 3 ⋅ 5k
⋅ 5 5
= 14 ⋅ 7k
+ 15 ⋅ 5k
5
= (12 + 2) ⋅ 7k
+ (12 + 3) ⋅ 5k
5
= 12 ⋅ 7k
+ 2 ⋅ 7k
+ 12 ⋅ 5k
+ 3 ⋅ 5k
5
= 12 ⋅ (7k
+ 5k
) + (2 ⋅⋅⋅⋅ 7k
+ 3 ⋅⋅⋅⋅ 5k
5)
F F 24 | (2 ⋅ 7k
+ 3 ⋅ 5k
5) F F 24 | 12 ⋅ (7k
+ 5k
)
ก F F F 24 | (2 ⋅ 7n
+ 3 ⋅ 5n
5) ก ก
n
ʿก 1.1
1. ก F a, b, c ˈ F ˈ F F F F
1) F a | b a | c F a | (b + c)
2) F a | b F am | bm m ∈ Z
3) F a | b F a | (b + c) F a | c
4) F a | b F a | (bx + cy) F a |c x, y ∈ Z
5) F b | c a | c
b F ab | c
6) F b | c c
b | a F c | ab
2. ก n F F F
1) 3 | (22n
1)
2) 2 | (n2
n)
3) 3 | (n3
n)

4 F
1.2 F ก
F 1.4 . . . 32 ก 48
ก ก 32 ก 96 F
32 = 2 ⋅⋅⋅⋅ 2 ⋅⋅⋅⋅ 2 ⋅⋅⋅⋅ 2 ⋅ 2
48 = 2 ⋅⋅⋅⋅ 2 ⋅⋅⋅⋅ 2 ⋅⋅⋅⋅ 2 ⋅ 3
F F ก F 32 ก 48 F กF 2 ⋅ 2 ⋅ 2 ⋅ 2 = 16
. . . 32 ก 48 F ก 16
ก . . . F 1.4 F ก ก ก F
ก 1, 2, 4, 8, 16, 32 32 F ( 32, 16, 8, 4, 2, 1) ก 1, 2, 3,
4, 6, 8, 12, 16, 24, 48 48 F ( 48, 24, 16, 12, 8, 6, 4, 3, 2, 1) ก F
ก ก F F ˈ ก 32 48 F F
F 32 48 F ก 16 ˈ ก F ˈ ก F F ก F
F F . . . 32 ก 48 F ก 16
F ก F ก ก F F
ก ˈ ʿก ก ก F F F F ก ก ก ก
F ก F
F ก d = (a, b) F F d | a d | b
m, k a = dm b = dk
a = dm = d( m) m ˈ
ก b = dk = d( k) k ˈ
1.2
ก F a, b ˈ F ˈ F ( F F F F ˈ F) F
ก F F d ∈ Z ˈ F ก (greatest common divisor : gcd) a, b ก F
1) d | a d | b
2) F c ∈ Z c | a c | b F c | d
1.3
ก F a, b ˈ F F F F ˈ F F d = (a, b) ˈ . . .
a, b F F F (a, b) = ( a, b) = (a, b) = ( a, b)

F F 5
d | ( a) d | ( b)
1.2.1 F F ( a, b) = ( a, b) = (a, b)
F ˈ ก F ก F ก
F ʽ F F F ก (Euclidean Algorithm) F
1.4 ก 1.5 ก F F ก ก F F F F F
F F F ก ก ก ก
F 1.5 . . . 128 ก 320 [(128, 320)] F x, y F
(128, 320) = 128x + 320y
กF F (128, 320) F 1.5
320 = 2 ⋅ 128 + 64
128 = 2 ⋅ 64
(128, 320) = 64
F x, y F 64 = 128x + 320y
1.4
ก F a, b ˈ F ˈ F F ก F d = (a, b) F x, y
d = ax + by
1.5 ก
ก a, b
F a = bq1 + r1 0 ≤ r1 ≤ b
b = r1q2 + r2 0 ≤ r2 ≤ r1
r1 = r2q3 + r3 0 ≤ r3 ≤ r2
⋮
rk 2 = rk 1qk + rk 0 ≤ rk ≤ rk 1
rk 1 = rkqk + 1
F rk = (a, b)

6 F
x, y ก F F F ก ก . . . ก
64 = 320 2 ⋅ 128
= 128( 2) + 320(1)
x, y F ก 2 ก 1
ʿก 1.2
1. . . . 24 ก 50 F F F F
2. F a, b, c, d ˈ d = (a, b) F F F
ก) ( )a b
d d, = 1
) (a + bc, b) = (a, b)
) F (a, b) = 1 a | bc F a | c
3. F d | (35n + 26), d | (7n + 3) d > 1 F d = 11
4. F a, b, m ˈ F ˈ F F (m, ab) = 1 ก F (m, a) = 1 = (m, b)
5. F F F a, b ˈ ก F F (a, b) = 2 ⋅ ( )a b
2 2,
6. F F F a ˈ F b ˈ F (a, b) = ( )a
2, b
7. F F F a, b ˈ (a, b) = 1 F (an
, bn
) = 1 ก
ก n
8. F F 7 F F F ก F 18,209 ก 19,043

F F 7
1.3 F F
F 1.6 [24, 40]
ก 24 = 23
⋅ 3
40 = 23
⋅ 5
F F ˈ ก F F กF 23
⋅ 3 ⋅ 5 = 120
[24, 40] = 120
F ˈ ก ก F F Fก F ก ˈ
ʿก
1.6 F F F F F F ก
( 1.5) F F F F F ก F ˈ ʿก
F ˈ F F . . . . . .
F F ก ก . . . . . . F ก . . .
. . . F
1.7 F F F F F F F F ก ก ก
ก
1.3
ก F a, b ˈ F ˈ F ( F F F F ˈ F) F
ก F F m ∈ Z ˈ F F (least common multiple : lcm) a, b ก F
1) a | m b | m
2) F c ∈ Z a | c b | c F m | c
1.6
F a, b, m ˈ a, b F ˈ F F ก F m = [a, b] ก F m > 0,
a | m, b | m m | n n ก ˈ F a b
1.7
ก F a, b ˈ a ⋅ b ≠ 0 F [a, b] = a b
(a, b)
⋅

8 F
ʿก 1.3
1. F 1.6
2. F . . . F F 1.6 ˈ
3. ก F a, b, c ˈ ก F [ca, cb] = c ⋅ [a, b]
4. F F a | b ก F [a, b] = | b |
5. ก F a, b ˈ ก F F [a, b] = a ⋅ b ก F (a, b) = 1
6. ก F a, b ˈ ก F (a, b) | [a, b]

2
2.1
F 2.1 F F ˈ F F ˈ
F ก ก F F
21, 37, 53, 69, 91, 111, 323, 301
ก 21 = 3 ⋅ 7 21 F ˈ
37 = 1 ⋅ 37 37 ˈ
53 = 1 ⋅ 53 53 ˈ
69 = 3 ⋅ 23 69 F ˈ
91 = 7 ⋅ 13 91 F ˈ
111 = 3 ⋅ 37 111 F ˈ
323 = 17 ⋅ 19 323 F ˈ
301 = 7 ⋅ 43 301 F ˈ
ก F 2.1 F F F ก F F F ก F F
F F F F ก ˈ F F ˈ F
ก ก กก F F ก 2 F ˈ Fก ˈ
F
F F F ก F
: n = 2 F F ˈ F 2 ˈ
: F k ˈ ก k ≥ 2
2.1
ก P F P 1 P F ก F F P ˈ
(Prime numbers) F P F ˈ F ก F F P ˈ ก (composite numbers)
2.1
ก n ≥ 2 F n ˈ Fก ˈ

10 F
F ˈ ก 2 < n ≤ k
F k + 1 ˈ F ˈ F F k + 1 F F
ก 2.1 F F k + 1 ˈ ก
p, q 2 < p ≤ k 2 < q ≤ k k + 1 = pq
F ก F F p, q ˈ
ˈ ก n = k + 1
ก F F F ˈ ก n ≥ 2
F F F F F ก n ก F ˈ
ก
F F n ˈ ก ˈ ก
a, b n = ab 1 < a ≤ b < n
F n = n ⋅ n F F a ≤ n
F a ˈ ก
2.1 F F a F ก F F
n F ˈ ก a F
ก n F F F F กก F n
F 2.2 F 1001 ˈ ก F ˈ ก F
F
ก 2.2 F 1001 = 31.64
ก F F ก 31.64 F กF 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31
F ก 11|1001 F 1001 F ˈ F F 1001 = 11 ⋅ 91
F F F F F
F k ( ก )
F 2, 3, 5, 7, 11, 13, , pk ˈ
F ก F n = (2 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 13 ⋅ ⋅ pk) + 1
2.2
ก n F n ˈ ก F n F ก F F กก F n
2.3
ก ˈ F

F F 11
F F (n 1)|(2 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 13 ⋅ ⋅ pk)
n F (2 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 13 ⋅ ⋅ pk)
F n F ˈ Fก ก F ก
F F F ก ˈ F F ก
ʿก 2.1
1. F F 161, 051 ˈ ก
2. F F ก p F p ˈ F p ˈ ก
2.2 ก ก
ก F ก F F F F
F F F ก ก ก ก
F 2.3 F 27 ก 35 ˈ Fก
1.4 F F F F
1 F F 1 = 27x + 35y x, y
35 = 27 ⋅ 1 + 8
27 = 8 ⋅ 3 + 3
8 = 3 ⋅ 2 + 2
3 = 2 ⋅ 1 + 1
2 = 1 ⋅ 1 + 1
F ก ก F F x, y F ก
ก 1 = 3 2 ⋅ 1
= 3 (8 3 ⋅ 2)
= (27 8 ⋅ 3) (8 3 ⋅ 2)
= (27 ⋅ 1 (35 27 ⋅ 1) ⋅ 3) ((35 27 ⋅ 1) (27 8 ⋅ 3) ⋅ 2)
2.2
ก a, b ˈ Fก (relatively prime) ก F (a, b) = 1

12 F
= 27 ⋅ 1 35 ⋅ 3 + 27 ⋅ 3 35 ⋅ 1 + 27 ⋅ 1 + 27 ⋅ 2 (35 27 ⋅ 1) ⋅ 3 ⋅ 2
= 27 ⋅ 7 35 ⋅ 4 35 ⋅ 6 + 27 ⋅ 6
= 27 ⋅ 13 + 35( 10)
x = 13, y = 10 F 1 = 27x + 35y
2.2 F F 27 ก 35 ˈ Fก F ก
F F a, b ˈ p ˈ
F p | ab ก 1.1 F F k ab = kp
p ˈ ก ab
F p | a F F ˈ
F F p Fa
F p ˈ 2.2 F F (a, p) = 1
ก ʿก 1.2 F 2 ( ) F p | b F ก
F F a, b ˈ d = (a, b)
F a = Md b = Nd
F d > 0 ( . . .) M = a
d N = b
d
F ก F ( a
d , b
d ) = 1
ก d = (a, b) 1.2 F F d |a d | b
1.1 p, q F a = pd b = qd
a
d = p ⋅ 1 b
d = q ⋅ 1
F F 1 |( )a
d 1 |( )b
d
1.2 F F ( a
d , b
d ) = 1
F M = a
d N = b
d 2.2 F F M, N ˈ Fก
F ก
2.4
F a, b ˈ p ˈ F p | ab F p | a p | b
2.5
F a, b ˈ d = (a, b) F a = Md b = Nd F M ก N ˈ
Fก

F F 13
ʽ F F 2 F ก F กF
ก (Fundamental Theorem of Arithmetic) F
2.6 F F F F F
F 2.4 ก ก 136
ก 136 = 8 ⋅ 17
F 8 F ˈ F 8 = 2 ⋅ 2 ⋅ 2 = 23
136 F F F 23
⋅ 17
F 2.5 (160, 224) [160, 224] F 2.6
ก 160 = 5 ⋅ 25
224 = 7 ⋅ 25
F 160 224 ก F 25
25
| 160 25
| 224
1.2 F F (160, 224) = 25
= 32
ก 5 ⋅ 7 ⋅ 25
= 1,120 ˈ F F F F
1.3 F F [160, 224] = 1,120
ʿก 2.2
1. F 147 ก 323 ˈ Fก F ก F 147 ก 323
2. F F F k ˈ ก F 3k + 2 5k + 3 ˈ Fก
2.6
F a ˈ ก กก F 1 F a F F F

3
ก F F
3.1 ก F F
F 3.1 ก F ก ˈ ก F
ก) 3x2
+ 2y2
= 5
) 4x + 3y = 1
) 3x3
y3
= 0
3.1 F F ก F ก) F ) F F F ˈ ก F
ก
ก) ก F 2 x, y ˈ F F
x = 1, y = 1
) ก F 2 x, y ˈ F F
x = 1, y = 1
) ก F 2 x, y ˈ F F
x = 0, y = 0
ก F F ˈ ก F กF F F
ก ก ก F F F (Non linear Diophantine Equation) F F F ก
ก ก ก
3.1
ก F (Diophantine Equation) ก ก F กก F
ˈ
3.2
ก F F (Linear Diophantine Equation) ก F F
a1x1 + a2x2 + a3x3 + + anxn = b a1, a2, a3, , an, b ˈ ai ≠ 0 ก 1 ≤ i ≤ n

16 F
F 3.2 ก F 3.1 ก F ˈ ก F F
ก 3.2 F ก ˈ ก F F F กF ก F )
ก x, y ˈ ˈ
ก F F ก 4x + 3y = 1 Fก F F F ก
ˈ F x = 1 y = 1 F F F F x = 2 y = 3
ก ˈ ก 4x + 3y = 1 ก F F ก F F F
กก F FกF ก ก ก ก F F
F ก ก ก F กF
ก F F F
F 3.3 ก 4x + 3y = 1
ก ก 4x + 3y = 1
กก x F F 3y = 1 4x
y = 1
3
4
3 x
F F x = t t ˈ
F y = 1
3
4
3 t ˈ ก F F ก F
F F t = 1 ก F F x = 1 y = 1 ˈ
ก F F ก F
3.3
x1, x2, x3, , xn F ก F F a1x1 + a2x2 + a3x3 + + anxn = b
F F ก ˈ ก F ก F F ( F ก F
)
1) ก ก F ˈ F
(general solution)
2) ก ก F F F กก F
F (particular solution)

F F 17
F 3.4 ก 2x + 6y = 7
ก ก 2x + 6y = 7
กก x F F 6y = 7 2x
y = 7
6
1
3x
F F F F x ˈ ก y ˈ F F
ก F ˈ
ก F F F F F ก ก F ก F F
ก ก ก ก F F ก F
F ก F ax + by = c F d = (a, b)
F ก F ก ˈ 3 F
F 1 F F dFc F ก F
ก F F F F F
F F ก F d |c
F ก F x0, y0 ˈ ก F
ax0 + by0 = c
F ก d = (a, b) d | a d | b
1.2 F F d | (ax0 + by0)
d | c F ก
F 2 F F d | c F ก
F d | c k c = dk
F ก d = (a, b) 1.4 F F p, q d = ap + bq
c = (ap + bq)k = a(pk) + b(qk)
F F x0 = pk y0 = qk ˈ ก
3.1
ก F ax + by = c ˈ ก F F F d = (a, b) F dFc
F ก F F F d | c F ก ˈ F F F
x0, y0 ˈ ก F ก F
x = x0 +( )b
d n y = y0 ( )a
d n n ˈ

18 F
F 3 F F ก F F F
x = x0 + ( )b
d n y = y0 + ( )a
d n n ˈ
F ก F x0, y0 ˈ ก
F F ax0 + by0 = c
(ax + by) (ax0 + by0) = c c = 0
F ax + by = ax0 + by0
a(x x0) = b(y0 y) -----(3.1.1)
ก d > 0 ( . . .)
ก (1) F d F
a
d (x x0) = b
d (y y0) -----(3.1.2)
1.1 F F b
d |( )a
d (x x0)
ก ʿก 1.2 F 2 (ก) ( ) F F b
d | (x x0)
1.1 F F x x0 = b
d n n
x = x0 + ( )b
d n -----(3.1.3)
F x ก ก (3) ก (2) F F
a
d ( )b
d n = b
d (y0 y)
y0 y = ( )a
d n
y = y0 ( )a
d n
F 3.5 ก 4x + 3y = 1
ก (3, 4) = 1 1|1
3.1 F ก ˈ F
ก F F x = x0 + ( )b
d n y = y0 ( )a
d n
ก F 3.1 F ก x0 = 1, y0 = 1
x = 1 + 3n y = 1 4n n ˈ
F 3.6 ก 12x + 28y = 4
ก (12, 28) = 4 4 | 4 ก
F F x = x0 + ( )b
d n y = y0 ( )a
d n
F x0, y0 F ก ( 1.5)

F F 19
28 = 2 ⋅ 12 + 4
12 = 3 ⋅ 4
4 = 28 2 ⋅ 12
= 12( 2) + 28(1)
F x0 = 2, y0 = 1
x = 2 + 7n y = 1 3n n ˈ
F 3.7 ก 121x + 33y = 99
ก (121, 33) = 11 11 | 99
ก F F x = x0 + ( )b
d n y = y0 ( )a
d n
ก F F
121 = 3 ⋅ 33 + 22
33 = 1 ⋅ 22 + 11
22 = 2 ⋅ 11
11 = 33 1 ⋅ 22
= 33 ⋅ 1 1 ⋅ (121 3 ⋅ 33)
= 4 ⋅ 33 1 ⋅ 121
ก F 9 F F 99 = 33(36) + 121( 9)
x0 = 9, y0 = 36
x = 9 + 3n y = 36 11n n ˈ
F 3.8 F A F B ก ก F ก 55 ก F
F F A F ก F 5 F F B ก
F ก F 6 F F ก F ก ก F A
F B
1) F A < F B
2) F A = F B
3) F A > F B
4) F F
F x = F A
y = F B

20 F
ก F A F B F ก 5x 6y
F F 5x + 6y = 55 -----(3.1.4)
3.2 F F ก (3.1.4) ˈ ก F F
ก (5, 6) = 1 1 | 55 3.1 ก (1) ˈ
F F x = x0 + ( )b
d n y = y0 ( )a
d n
ก 1 = 5 ⋅ ( 1) + 6 ⋅ 1 -----(3.1.5)
ก (3.1.5) F 55 F 55 = 5 ⋅ ( 55) + 6 ⋅ 55
ก x0 = 55, y0 = 55
F ก x = 55 + 6n y = 55 5n n ˈ
ก F F F ก ˈ ก F
x ≥ 0 y ≥ 0
F F 55 + 6n ≥ 0 55 5n ≥ 0
n ≥ 9 n ≤ 11
F n F ก n = 10
F x = 55 + 6(10) = 55 + 60 = 5
y = 55 5(10) = 5
F F F A = F B
กF ˆ F 3.8 F ก ก กF ˆ F 3.3 ก F
ก F F ก
ก 5x + 6y = 55
กก x F 6y = 55 5x
y = 55
6
5
6 x
F x = 5 F F y = 5 > 0
F ก F F ˈ ก ก F F กก
F ก F F Fก ก F F

F F 21
F 3.9 ก 2x + 3y + 4z = 6
F 2x + 3y + 4z = 6
(2x + 4z) 6 = 3y = 3( y)
F F 3 | ((2x + 4z) 6) (2x + 4z) ≡ 6 (mod 3)
F 6 ≡ 0 (mod 3) F F (2x + 4z) ≡ 0 (mod 3)
ก 2x ≡ 4z (mod 3)
ก 2 ≡ 1 (mod 3) F 2x ≡ x (mod 3)
F F x ≡ 4z (mod 3)
4.3 F F x ≡ 4z (mod 1)
F ก F z = m ∈ Z F F x = 4m + n n
F x, z ก F F 3y = 6 4m 2(4m + n) = 6 12m 2n
y = 2 4m 2
3 n
F ก F ก ˈ
F F n = 3k k ก F F y = 2 4m 2k
F ก
x = 4m + 3k,
y = 2 4m 2k,
z = m m, k ∈ Z
ʿก 3.1
1. F ก F F ก F F F F
F
1) 17x + 13y = 100
2) 1402x + 1969y = 1
3) 101x + 102y + 103z = 1

4
ก ก ก ก F
4.1 ก (modulo)
กF ก ก ก ก ก ก F ก F
ก F ˈ ก ก F ก
ก F (modulo) F
F 4.1 F F
1) 16 (mod 3)
2) 15 (mod 3)
3) 16 (mod 3)
4) 15 (mod 3)
5) 0 (mod 6)
6) 1 (mod 6)
1) ก 16 = 5(3) + 1
16 (mod 3) = 1
2) ก 15 = 5(3) + 0
16 (mod 3) = 0
3) ก 16 = 5(3) 1
16 (mod 3) = 1
4) ก 15 = 5(3) + 0
15 (mod 3) = 0
5) ก 0 = 0(6) + 0
0 (mod 6) = 0
6) ก 1 = 0(6) 1
1 (mod 6) = 1
4.1 ก (modulo: mod)
ก F a, m ˈ m > 0 F a (mod m) F กก a F m
ก m F (modulus)

24 F
ʿก 4.1
1. F 6 ก 20 ก
2. F 7 ก 20 ก
3. F 21 ก F 100 120
4. F 19 ก F 171 200
5. ก F F 31 ก 100 ก
4.2 ก ก (congruence)
F 4.2 ก F 4.1 F F a ≡ b (mod m)
1) ก 3 | (16 1) 16 ≡ 1 (mod 3)
2) ก 3 | (15 0) 15 ≡ 0 (mod 3)
3) ก 3 | ( 16 ( 1)) 16 ≡ 1 (mod 3)
F 4) 6) F F F ˈ ʿก
F ˈ F ก ก ก ก F Fก F F ʿก
F F F F F F F ก ก ก ก
F F a, b, m ˈ m > 0
(ﬂ) F a ≡ b (mod m)
4.2 F F m | (a b)
1.1 F F a b = km k
a = b + km -----(4.2.1)
1.5 q, r F b = qm + r -----(4.2.2)
4.2
ก F a, b, m ˈ m > 0 F a ก b m ก F m | (a b)
F ก F a ≡ b (mod m) a ก b m
4.1
ก F a, b, m ˈ m > 0 F a ≡ b (mod m) ก F a (mod m) = b (mod m)

F F 25
F m b F ก r
F F ก (4.2.2) ก (4.2.1) F F
a = (qm + r) + km
= (q + k)m + r
F m a F ก r F ก
a (mod m) = b (mod m) F ก
(›) F a (mod m) = b (mod m)
1.5 F F q1, q2, r1, r2 a = mq1 + r1 b = mq2 + r2
0 ≤ r1 ≤ m 0 ≤ r2 ≤ m
F F r1 = a mq1 r2 = b mq2
F ก F F r1 = r2
a mq1 = b mq2
a b = mq1 mq2 = m(q1 q2)
1.1 F F m | (a b)
4.2 F a ≡ b (mod m) F ก
F ก F a, b, c, d, m ˈ m > 0, a ≡ b (mod m) c ≡ d (mod m)
1) ก a ≡ b (mod m) c ≡ d (mod m)
4.2 F F m | (a b) m | (c d)
1.1 F F a b = k1m c d = k2m k1, k2
a = b + k1m -----(4.2.1)
c = d + k2m -----(4.2.2)
ก (4.2.1) + ก (4.2.2); a + c = (b + k1m) + (d + k2m)
a + c = (b + d) + (k1 + k2)m
F F m | [(a + c) (b + d)] 4.2 F F a + c ≡ b + d (mod m)
2) ก F 1) F a = b + k1m c = d + k2m k1, k2
F F ac = (b + k1m)( d + k2m) = bd + dk1m + bk2m + k1k2m2
4.2
ก F a, b, c, d, m ˈ m > 0, a ≡ b (mod m) c ≡ d (mod m) F
F F ˈ
1) a + c ≡ b + d (mod m)
2) ac ≡ bd (mod m)

26 F
= bd + (dk1 + bk2 + k1k2m)m
m | (ac bd) 4.2 F F ac ≡ bd (mod m)
F ก F a, b, c, m ˈ m > 0
(ﬂ) F F ca ≡ cb (mod m) F a ≡ b( )m
(c, m)mod
F ca ≡ cb (mod m) F d = (c, m)
4.2 F F m |(ca cb)
1.1 F F ca cb = c(a b) = qm q -----(4.2.3)
ก d > 0 ก (4.2.3)
F F c
d (a b) = ( )m
dq 1.1 F F m
d | c
d (a - b)
F ( )c m
d d, = 1 ( ʿก 1.2 F 2 (ก)) m
d | (a b)
4.2 F F a ≡ b( )m
(c, m)mod
(›) F F F F F ˈ ʿก
ca ≡ cb (mod m) ก F a ≡ b( )m
(c, m)mod F ก
F ก F a, b ˈ F m1, m2 ˈ ก (m1, m2) = 1
(ﬂ) F a ≡ b (mod m1m2)
4.2 F F m1m2 | (a b)
1.1 F F k a b = km1m2
F m1, m2 ˈ Fก (m1, m2) = 1
1.2 F F 1 | m1 1 | m2
1.1 F F m1 = k1 m2 = k2 k1, k2
F F a b = (kk1)m2 m2 | (a b)
ก a b = (kk2)m1 m1 | (a b)
4.3
ก F a, b, c, m ˈ m > 0 F F
ca ≡ cb (mod m) ก F a ≡ b( )m
(c, m)mod
4.4
ก F a, b ˈ F m1, m2 ˈ ก m1, m2 ˈ F
ก F a ≡ b (mod m1m2) ก F a ≡ b (mod m1) a ≡ b (mod m2)

F F 27
F F a ≡ b (mod m1) a ≡ b (mod m2) F ก
(›) F a ≡ b (mod m1) a ≡ b (mod m2)
4.2 F F m1 | (a b) m2 | (a b)
1.1 F F a b = qm1 a b = rm2 q, r
F m1, m2 ˈ Fก
s a b = sm1m2
1.1 F F m1m2 | (a b)
a ≡ b (mod m1m2) F ก
ʿก 4.2
1. ก F a, b, c, m ˈ m > 0 F F F
ก) F ca ≡ cb (mod m) (c, m) = 1 F a ≡ b (mod m)
) F ca ≡ cb (mod m) c > 0 c | m F a ≡ b( )m
cmod
2. ก F a, b ˈ m, n ˈ ก a ≡ b (mod m) F F F n | m
F a ≡ b (mod n)
3. ก F a, b ˈ m, c ˈ ก F F F a ≡ b (mod m) F
ca ≡ cb (mod cm)
4. ก F a, b, m ˈ m > 0 a ≡ b (mod m) F F (a, m) = (b, m)
5. F 4.3 F F

28 F
4.3 ก ก
F F F ก ก ก ก ก ก ก ก ก
F F ก ก ก ก F กF ˆ
F F
F F a, b, c, m ˈ m > 0 a ≡ b (mod m)
1) ก a ≡ b (mod m) 4.2 F F m | (a b)
1.1 k a b = km
a = b + km -----(4.3.1)
ก ก (4.3.1) F c
F a + c = (b + km) + c = (b + c) + km
(a + c) (b + c) = km
1.1 F F m | [(a + c) (b + c)]
F a + c ≡ b + c (mod m) F ก
2) ก ก (4.3.1) ก F c
F ac = (b + km)c = bc + (kc)m
ac bc = (kc)m
1.1 F F m | (ac bc)
F ac ≡ bc (mod m) F ก
3) F F ก F
F P(n) : an
≡ bn
(mod m) ก n
: n = 1; ก a ≡ b (mod m) P(1) ˈ
: F n = k; F P(k) ˈ F P(k + 1) ˈ
ก ak
≡ bk
(mod m) ก a ≡ b (mod m)
4.2 (2) F F ak
⋅ a ≡ bk
⋅ b (mod m)
ak + 1
≡ bk + 1
(mod m) F P(k + 1) ˈ
4.5
ก F a, b, c, m ˈ m > 0 F a ≡ b (mod m) F F
1) a + c ≡ b + c (mod m)
2) ac ≡ bc (mod m)
3) ก n F F an
≡ bn
(mod m)

F F 29
ก F F F an
≡ bn
(mod m) ก n
F 4.3 F F F
1) F a ˈ F a2
≡ 1 (mod 8)
2) F a ˈ F F a2
≡ 0 (mod 4)
1) F a ˈ
F F a = 2k + 1 k
a2
= (2k + 1)2
= 4k2
+ 4k + 1
= 4(k2
+ k) + 1
= 4(k(k + 1)) + 1
ก k ˈ F k ก k + 1 ˈ F
m F k(k + 1) = 2m
F F a2
= 4(2m) + 1 = 8m + 1
a2
≡ 1 (mod 8)
2) F a ˈ F
F F a = 2k k
a2
= (2k)2
= 4k2
F F 4|a2
a2
≡ 0 (mod 4)
F 4.4 กก F F F
1) 250
F 7
2) 710
F 51
3) 521
F 127
4) 15
+ 25
+ + 105
F 4
1) ก 250
= (23
)16
⋅ 22
F 250
≡ (23
)16
⋅ 22
(mod 7)
23
≡ 1 (mod 7) 22
≡ 4 (mod 7)
250
≡ 116
⋅ 4 (mod 7) ≡ 4 (mod 7)
250
F 7 F ก 4
2) ก 710
= (74
)2
⋅ 72
F F 710
≡ (74
)2
⋅ 72
(mod 51)

30 F
74
≡ 4 (mod 51) 72
≡ 2 (mod 51)
710
≡ 42
⋅ ( 2) (mod 51) ≡ 42
⋅ 72
(mod 51)
≡ 19 (mod 51)
710
F 51 F ก 19
3) ก 521
= (56
)3
⋅ 53
F F 521
≡ (56
)3
⋅ 53
(mod 127)
56
≡ 4 (mod 127) 53
≡ 53
(mod 127)
521
≡ 43
⋅ 53
(mod 127)
≡ 203
(mod 127)
≡ 126 (mod 127)
521
F 127 F ก 126
4) ก 1 ≡ 1 (mod 4) ≡ 5 (mod 4) ≡ 9 (mod 4)
2 ≡ 2 (mod 4) ≡ 6 (mod 4) ≡ 10 (mod 4)
3 ≡ 3 (mod 4) ≡ 7 (mod 4)
4 ≡ 0 (mod 4) ≡ 8 (mod 4)
15
≡ 1 (mod 4) ≡ 55
(mod 4) ≡ 95
(mod 4)
25
≡ 25
(mod 4) ≡ 65
(mod 4) ≡ 105
(mod 4)
35
≡ 35
(mod 4) ≡ 75
(mod 4)
45
≡ 05
(mod 4) ≡ 85
(mod 4)
15
+ 25
+ 35
+ + 105
≡ 3(15
) + 3(25
) + 2(35
) + 2(05
) (mod 4)
≡ 3 + 96 + 486 + 0 (mod 4)
≡ 3 + 0 + 2 (mod 4)
≡ 5 (mod 4)
≡ 1 (mod 4)
F F 15
+ 25
+ + 105
F 4 F ก 1
F 4.5 F F ก ก F
1) 223
1 F 47
2) 248
1 F 97
1) ก
212
≡ 7 (mod 47)
210
≡ 37 (mod 47)

F F 31
2 ≡ 2 (mod 47)
223
≡ 212
⋅ 210
⋅ 2 (mod 47)
≡ 7 ⋅ 37 ⋅ 2 (mod 47)
≡ 518 (mod 47)
≡ 1 (mod 47)
223
1 ≡ 1 1 (mod 47) ≡ 0 (mod 47)
223
1 F 47
2) ก
212
≡ 22 (mod 97)
248
≡ (212
)4
(mod 97)
≡ 224
(mod 97)
≡ 70 ⋅ 16 (mod 97)
≡ 1 (mod 97)
248
1 ≡ 1 1 (mod 97) ≡ 0 (mod 97)
248
1 F 97
F 4.6 ก F a, b ˈ p ˈ ก F F F a2
≡ b2
(mod p)
F a ≡ ± b (mod p)
F a, b ˈ p ˈ ก
F a2
≡ b2
(mod p)
F F p | (a2
b2
)
F a2
b2
= (a b)(a + b)
p |(a b)(a + b)
2.4 F F p |(a b) p |(a + b)
a ≡ b (mod p) a ≡ b (mod p)
a ≡ ± b (mod p)
ʿก 4.3
1. ก F a, b ˈ p ˈ ก
1) F F 1 + 2 + 3 + + (n 1) ≡ 0 (mod n) ก F n ˈ
2) F F F a2
≡ a (mod p) F a ≡ 0 (mod p) a ≡ 1 (mod p)

32 F
2. กก F F
1) 13
+ 23
+ 33
+ + 103
F 11
2) 1 + 3 + 5 + 7 + + 101 F 7
3) 12
+ 32
+ 52
+ 72
+ + 1012
F 131
4.4 ก ก F
4.4.1 ก ก ก F
F 4.7 F ก ก ก F F ˈ ก ก F F
1) 3x ≡ 1 (mod 4)
2) x ≡ 2 (mod 5)
3) 2x2
≡ 3 (mod 7)
4) x3
≡ 1 (mod 13)
4.3 F F ก ก ก F F 3) F 4) F Fก ก
F F ก x F F ก 1 F ก ก ก F F
ˈ ก ก F
4.3
ก F a, b, m ˈ m > 0 กก ก F ax ≡ b (mod m) F
ก ก F (linear congruence) x ˈ

F F 33
F ก ก F F F F 3 ก ก F
F ก F ก F FกF F
ก ก F กF
F ก ก F ก
ก F F ˈ F F F ก ก F ก
F F
F ก F ax ≡ b (mod m) ˈ ก ก F m F d = (a, m)
(ﬂ) F F ax ≡ b (mod m) ˈ F d | b
F ax ≡ b (mod m) ˈ
F x0 ˈ ก ก F F F ax0 ≡ b (mod m)
4.2 F F m | (ax0 b)
1.1 F F ax0 b = mk k
b = ax0 mk
ก d = (a, m) 1.2 F F d | a d | m
1.2 F F d | (ax0 mk)
F F d | b F ก
(›) F F d | b F ก ก ax ≡ b (mod m) ˈ
F d | b
1.1 r F b = rd
ก d = (a, m) 1.2 x1, y1 F d = ax1 + my1
ก F r F rd = r(ax1) + r(my1) = a(rx1) + m(ry1)
4.4
ก F ax ≡ b (mod m) ˈ ก ก F ก x0 F ax0 ≡ b (mod m)
F ก ก ax ≡ b (mod m) F x1 ก F
ax1 ≡ b (mod m) ˈ ก x0 ≡ x1 (mod m) F ก x0, x1 F ก
(congruent solution) F F x0 T x1 (mod m) F ก x0, x1 F F ก (incongruent
solution)
4.6
ก F ax ≡ b (mod m) ˈ ก ก F m F d = (a, m) F ก ก
ax ≡ b (mod m) ˈ ก F d | b

34 F
F b = rd b = a(rx1) + m(ry1)
F a(rx1) b = m(ry1) = m( ry1)
m | [a(rx1) b]
a(rx1) ≡ b (mod m)
F ก ก ax ≡ b (mod m) rx1 ˈ
F 4.8 F ก ก F ก F F F
1) 2x ≡ 1 (mod 3)
2) 3x ≡ 3 (mod 4)
3) 4x ≡ 4 (mod 18)
1) ก 2x ≡ 1 (mod 3)
ก (2, 3) = 1 1 | 1 ก ก ˈ F
2) ก 3x ≡ 3 (mod 4)
ก (3, 4) = 1 1 | 3 ก ก ˈ F
3) ก 4x ≡ 4 (mod 18)
ก (4, 18) = 2 2 | 4 ก ก ˈ F
F 4.4 F F F F ก F F ก ก ก F
F ก กก F F F F F F ก ก F
ˈ 4.6 ก F ก
F ก F 4.7 F F ก F ก F
F 4.9 ˈ F ก 2x ≡ 1 (mod 3)
ก F 4.8 (1) F ก ก ก F ˈ F
ก d = 1 m = 3 F F x = x0 + 3n n = 0 x = x0
ก 2x ≡ 1 (mod 3) F F 2x 3y = 1 y -----(4.4.1)
4.7
ก F ax ≡ b (mod m) ˈ ก ก F m d = (a, m) F d | b F ก
ก ax ≡ b (mod m) F ก m d
x = x0 + ( )m
d n n = 0, 1, 2, , d 1 x0 ˈ ก ก
ax ≡ b (mod m)

F F 35
ก 1 = 2( 1) 3( 1)
x0 = x = 1 ≡ 1 (mod 3) ˈ ก ก ก F
F 4.10 ˈ F ก 4x ≡ 4 (mod 18)
ก F 4.8 (3) F F d = 2 ก ก F ก 2
m = 18 F F x = x0 + 9n n = 0, 1
ก 4x ≡ 4 (mod 18) F F 4x 18y = 4
18 = 4 ⋅ 4 + 2
4 = 2 ⋅ 2
F F 2 = 18 ⋅ 1 4 ⋅ 4
4 = 18 ⋅ 2 4 ⋅ 8
= 4( 8) 18( 2)
F F x0 = 8 ˈ ก ก ก F
F ก 2 F ก
x1 = 8 + 9(0) = 8 ≡ 8 (mod 18) ≡ 10 (mod 18)
x2 = 8 + 9(1) = 1 ≡ 1 (mod 18)
F 4.11 ก ก 15x ≡ 3 (mod 9)
ก 15 ≡ 6 (mod 9)
F 15x ≡ 6x ≡ 3 (mod 9)
ก (6, 9) = 3
F F 2x ≡ 1 (mod 3)
ก 2 ≡ 2( 1) ≡ 1 (mod 3)
F x0 = 1 ˈ ก ก 2x ≡ 1 (mod 3) ˈ ก
ก 15x ≡ 3 (mod 9) F
F x = 1 + 3n n = 0, 1, 2
n = 0; x = 1 ≡ 8 (mod 9)
n = 1; x = 1 + 3(1) = 2 ≡ 2 (mod 9)
n = 2; x = 1 + 3(2) = 5 ≡ 5 (mod 9)
ก ก x ≡ 2 (mod 9), x ≡ 5 (mod 9), x ≡ 8 (mod 9)
3 F (6, 9) = 3

36 F
4.4.2 ก ก ก (inverse of congruence)
F 4.12 ก 7 17
ก ก 7x ≡ 1 (mod 17)
F F 7x 17y = 1 -----(4.4.2)
ก 1.5 F
17 = 7(2) + 3
7 = 3(2) + 1
(7, 17) = 1 F F ก ก 1 F
ก 1 = 7 3(2)
= 7 2(17 7(2))
= 7(1) 2(17) + 7(4)
= 7(5) 17(2)
x = x0 = 5
F F 5 ˈ ก 7 (mod 17)
F ก ก ก ก ก ก ก ก F ˈ ก
ก ก F ก ก ก F F 4.4.1 F
F 4.13 ก ก 7x ≡ 7 (mod 17)
ก 7x ≡ 7 (mod 17)
ก ก F 5 ( ก F 4.12 F ก 7 17 5)
F 5 ⋅ 7x ≡ 5 ⋅ 7 (mod 17) ≡ 35 (mod 17)
F 5 ⋅ 7x ≡ x ≡ 35 (mod 17)
≡ 1 (mod 17)
ก ก x ≡ 1 (mod 17) F (7, 17) = 1
4.4
ก F a, m ˈ m > 0 (a, m) = 1 F a ˈ ก ก
ax ≡ 1 (mod m) F ก a F ก a m

F F 37
ʿก 4.4
1. ก ก F F
1) 3x ≡ 2 (mod 7)
2) 6x ≡ 3 (mod 9)
3) 5x ≡ 6 (mod 17)
4) 4x ≡ 12 (mod 17)
5) 623x ≡ 511 (mod 679)
6) 481x ≡ 627 (mod 703)
2. ก ก 1333 1517
3. ก ก F F
1) 2x + 3y ≡ 4 (mod 7)
2) 3x + 6y ≡ 2 (mod 9)
3) 8x + 2y ≡ 4 (mod 10)
4. ก F a, b ˈ a′ ˈ ก ก a m b′ ˈ ก ก
b m F a′b′ ˈ ก ก ab m

F F 39
ก
. F Ent 45. ก : ก , 2545.
F. . ก : F, 2543.
ก . F . F 2. ก : F ก F, 2546.
F . 1. ก : , 2545.
ก . ก F .4 ( 011, 012). ก : ʽ ก F F, 2539.

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