Chinese Remainder Theorem 中国剩余定理

中国剩余定理CRT (Chinese Remainder Theorem)
X ≡ 2 (mod 3)
X ≡ 3 (mod 5)
X ≡ 2 (mod 7)
Solve X?
明. 程大位 "算法统宗" (1593)
3人同行70稀
5树梅花21支
7子团圆半个月(15)
除百零五(105)便得知
Let remainders:
$Latex r_3=2, r_5=3, r_7=2$
$Latex r= r_3.70+ r_5.21 + r_7.15 (mod 3.5.7)$
r= 2.70 +3.21 +2.15 (mod 105)
r= 140 +63 +30 (mod 105)
r= 233 (mod 105)
$latex r= 23 = x_{min}$
or X= 23 +105Z (23 + multiples of 105)

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CRT: Why 3:70, 5:21, 7:15
X ≡ 2 (mod 3)
X ≡ 3 (mod 5)
X ≡ 2 (mod 7)

1) Find A such that
A ≡ 1 (mod 3)
A ≡ 0 (mod 5)
A ≡ 0 (mod 7)

=> 5|A, 7|A => 35 |A

A=35, 70 ...

70 ≡ 1 (mod 3)

=> 70x2 ≡ 2 (mod 3)

2) Find B s.t.:

B ≡ 0 (mod 3)

B ≡ 1 (mod 5)

B ≡ 0 (mod 7)

3|B, 7|B => 21|B

21 ≡ 1 (mod 5)

=> 21x3 ≡ 3 (mod 5)

3) Find C s.t. :

C ≡ 0 (mod 3)

C ≡ 0 (mod 5)

C ≡ 1 (mod 7)

=> 3|C, 5|C => 15|C

=> C=15≡ 1 (mod 7)

15x2 ≡ 2 (mod 7)

4)

X ≡ 70x2 +21x3+ 15x2 (mod 3x5x7)

X≡ 233 (mod 105)

X≡ 23 (mod 105)

X= 23+105Z

----------------Ring Theory -------------------------------
Commutative Ring CRT by Ring/ Ideal Theory:
If R is a commutative ring & A1,...An are pairwise coprime ideals,
Prove that if
$Latex r_1, r_2, \cdots, r_n$ belong to R
Then there exists 'a' belongs to R with
$Latex a + A_j= r_j + A_j$ ; j ∈[1,n]
Interpretation:
Let R= Z ring
X ≡ 2 (mod 3)
X ≡ 3 (mod 5)
X ≡ 2 (mod 7)
Or
X ≡ rj (mod mj)

Ideal = (m) = mZ
$Latex A_3=(3), A_5=(5), A_7=(7)$
$latex r_3=2, r_5=3, r_7=2$
There exists
a=23
$Latex a +A_3 = r_3 + A_3$
23+(3)= 2+ (3)
23 + 1x3 =2 + 8x3 =26