CONGRUENCES

m
a b m b −a
m
a≡b(mod.m)a≡b[m]
43 ≡83 (mod.10) 83 −43 = 40
351 ≡843 (mod.41) 843 −351 = 492
55 ≡ −37 (mod.23) −37 −55 = −92
a≡b[m]a b
m
a≡b[m]k b −a=km
a m a =mq +r0≤r < m
b=m(k+q) + r0≤r < m
b m r
a b r m
a=mq +r b =mq0+r b −a=m(q−q0)b−a
m a ≡b[m]
a b c d m
a≡a[m]
a≡b[m]b≡a[m]
a≡b[m]b≡c[m]a≡c[m]
a≡b[m]c≡d[m]a+c≡b+d[m]
a≡b[m]c≡d[m]ac ≡bd [m]
a≡b[m]ap≡bp[m]p≥1)
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bd −ac =d(b−a) + a(d−c)
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n n6−1

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10 ≡3 [7] ,100 ≡2 [7] ,1000 ≡ −1 [7] ,10 000 ≡ −3 [7] ,100 000 ≡ −2 [7] ,
1 000 000 ≡1 [7]
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73n−1n
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33≡1 [13] 33n≡1 [13]
36n+2 + 33n+1 + 1
n
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32n+1 + 2n+2 ≡0 [7] n
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9≡ −1 [10] 92n92n+1
n
99 ≡ −1 [100] 992n
992n+1 n
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3245
57 383114
91 2341 998
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a b n
a−b an−bnn a +b an+bn
a > b a ≡b[a−b]

n=cp10p+cp−110p−1+. . . +c110 + c0
10 ≡1 [9] ,10k≡1 [9] k n ≡cp+cp−1+. . .+c1+c0[9]
n≡0 [9]
cp+cp−1+. . . +c1+c0≡0 [9]
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x
3x79
12345x
351x4
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x y
43xy
x431y
28x75y