n
m " N, a, b " Z
m|a - b a a"m b
a a"m b
5 a"3 2 3|(5 - 2) 5 a"3 3 3 (5 - 3)
m " N m Z
a b m
a a" b (mod m), a = b (mod m), a =m b.
(a a"m b) Ð!Ò! (m|a - b) Ð!Ò! ("k a - b = mk) Ð!Ò! ("k a = b + mk) .
a, b " Z, m " N a a"m b
a b m
a, b " Z m " N
Ò! a a"m b " q1, r1, q2, r2 "
Z
a = q1m + r1, b = q2m + r2,
0 d" r1, r2 < m (")
(a - b) = (q1 - q2)m + (r1 - r2).
m | (a - b)
m | r1 - r2 " k km = r1 - r2 r1 = km + r2 (")
k = 0 r1 = r2
Ð! "k, t, r " Z, 0 d" r < m a = km + r, b = tm + r
a - b = m(k - t) a a"m b.
m
Z
 a a"m a m|a - a
a a"m b b a"m a m|a - b m|b - a
a a"m b b a"m c a a"m c
a - c = (a - b) + (b - c).
a, b, c, d " Z, m, mi, k " N
a a"m b Ð!Ò! ak a"mk bk
ad a"m bd '" NW D(d, m) = 1 =Ò! a a"m b
a a"m b =Ò! NW D(a, m) = NW D(b, m)
a a"m b, a a"m b NW D(m1, m2) = 1 a a"m m2 b
1 2 1
NW D(a, m) = 1, NW D(b, m) = 1 =Ò! NW D(ab, m) = 1.
m|a - b Ð!Ò! mk|ak - bk
m|(a - b)d, NW D(m, d) = 1 =Ò! m|a - b
m|a - b =Ò! NW D(a, m) = NW D(b, m) "k a = km + b
m1|a - b, m2|a - b, NW D(m1, m2) = 1
NW D(ab, m)
a a"m b NW D(a, m) =
1 NW D(b, m) = 1 [a]
m
Z
m.
m
{0, 1, ..., m - 1} m
m
{0, 1, ..., m-1}
m
 m " N
m
a [a]m
m
m
0, 1, ..., m - 1 m
r a"m r - m m
m - 1 m - 1 m - 1
- , - + 1, ..., -1, 0, 1, ..., ,
2 2 2
m = 7 -3, -2, -1, 0, 1, 2, 3.
m
m m m
- , - + 1, ..., -1, 0, 1, ..., - 1,
2 2 2
m = 8 -4, -3, -2, -1, 0, 1, 2, 3.
m m - 1
- , ..., -1, 0, 1, ..., ,
2 2
x x
a " R
a = min{x ; ], x " Z, x e" a}, a = max{x ; x " Z, x d" a}.
2k-1 1 8-1
= k - = k - 1 -7 = -4, = 3.
2 2 2 2
a, b " Z, m " N X = {x1, x2, ..., xm}
m NW D(a, m) = 1
aX + b := {axi + b, xi " X}
m
a, b " Z, m " N X = {x1, x2, ..., xm}
m aX + b m
m
m
" i = j axi + b a"m axj + b b a"m b

ax1 = ax1 + b - b a"m ax2 + b - b = ax2.
(a, m) = 1 x1 a"m x2 x1, x2
X
 n
n Õ(n)
m " N m
m
m
Õ(m) m
m m
a a"m b
NW D(a, m) = 1 NW D(b, m) = 1
m
m
m
{0, 1, ..., m - 1}
m
m Õ(m) Õ
10 Õ(10) = Õ(2)Õ(5) =
4 {1, 3, 7, 9}.
m " N, a " Z
NW D(a, m) = 1 X = {x1, x2, ..., xÕ(m)}
m aX
" i = j axi a"m axj NW D(axi, m) = 1

" i = j axi a"m axj NW D(a, m) = 1

x1 a"m x2 X
" i = j axi a"m axj

NW D(a, m) = 1 NW D(xi, m) = 1
" i NW D(axi, m) = 1.
m Z
"a, b, c, d " Z m " N a a"m b c a"m d
(a + c) a"m (b + d)
(ac) a"m (bd)
a a"m b, c a"m d, m|(a - b) i m|(c - d).
m|(a - b) + (c - d) =Ò! m|(a + c) - (b + d) =Ò! (a + c) a"m (b + d),
m|c(a - b) + b(c - d) =Ò! m|ac - bd =Ò! (ac) a"m (bd).
Z
 a + c a"m b + d
ac a"m bd
a, b " Z m, n " N
a a"m b, c a"m d a - c a"m b - d
a a"m b " k " N ak a"m bk.
n n
" i = 1, ..., n ai, bi " Z ai a"m bi ai a"m bi
i=1 i=1
m m, n "
N W (x1, x2, ..., xn) n
1 n
W (x1, ..., xn) = a(Ä… ,...,Ä…n)xÄ… · · · xÄ… ,
1 1 n
(Ä…1
,...,Ä…n
)
Ä…i e" 0, a(Ä… ,...,Ä…n) " Z.
1
" i = 1, 2, ..., n ci a"m di W (c1, ..., cn) a"m W (d1, ..., dn).
(Ä…1, ..., Ä…n) a(Ä… ,...,Ä…n) a"m b(Ä… ,...,Ä…n)
1 1
n
1 n 1
a(Ä… ,...,Ä…n)cÄ… · · · cÄ… a"m b(Ä… ,...,Ä…n)dÄ… · · · dÄ… .
1 1 n 1 1 n
(Ä…1
,...,Ä…n
) (Ä…1
,...,Ä…n
)
W (x1, x2, x3) = 5x2 + 3x1x3 - 8x1x2
1
5 a"3 2, 1 a"3 1, 2 a"3 2 W (5, 1, 2) a"3 W (2, 1, 2)
3 a"3 0, -8 a"3 1,
W (5, 1, 2) a"3 W (2, 1, 0) a"3 2 · 22 + 0 · 2 · 1 + 1 · 2 · 1 = 8 + 2 = 10 a"3 1.
3, 9, 11
a
a = an10n + ... + a110 + a0, , n " N, " i = 0, 1, ..., n ai " {0, 1, ..., 9}.
W (x) = anxn + an-1xn-1 + ... + a1x + a0
ż2
 W (10) = a 10 a"3 1 10 a"9 1
a = W (10) a"3 W (1) = an + an-1 + ... + a1 + a0
a = W (10) a"9 W (1) = an + an-1 + ... + a1 + a0.
10 a"11 -1
a a"11 a0 - a1 + a2 - a3 + · · · + (-1)nan
a