chinese remainder theorem. how many people what is x? divided into 4s: remainder 3 x ≡ 3 (mod 4)...
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Chinese Remainder Theorem
How many peopleWhat is x?
Divided into 4s: remainder 3x ≡ 3 (mod 4)
Divided into 5s: remainder 4x ≡ 4 (mod 5)
Chinese Remainder Theorem
Chinese Remainder Theorem
x ≡ a1 (mod m1)
x ≡ a2 (mod m2)
x ≡ ak (mod mk)
Theorem: If m1,m2,…,mk are relatively prime and
a1,a2,…,ak are integers, then
have a unique solution modulo m, where m = m1m2…mk.
(That is, there is a solution x with and all other solutions are congruent modulo m to this solution.)
mx 0
(1) Compute m = m1 m2 … mn .
(2) Determine M1 = m/m1 ; M2= m/m2 ; … ; Mn= m/mn
(3) Find the inverse of M1 mod m1, M2 mod m2 , …, Mn
mod mn which are y1 , y2 ,…, yn ,
(4) Compute x = a1 M1 y1 + a2 M2 y2 +…+ an Mn yn
(5) Solve x ≡ y (mod m)
Steps of solution:
Example : Solve the system of congruences
x≡ 2 (mod 3), x≡ 3 (mod 5), x≡ 2 (mod 7)
Solution:
(1) m= 3 · 5 · 7=105(2) M1 = m/m1 =105/3=35, M2 = 21; M3 = 15(3) y1 = 2 is an inverse of 35 mod 3 because 35 ≡ 2 (mod 3)
y2 = 1 is an inverse of 21 mod 5 because 21 ≡ 1 (mod 5)
y3 = 1 is an inverse of 15 mod 7 because 15 ≡ 1 (mod 7) (4) x= a1 M1 y1 + a2 M2 y2 + a3 M3 y3 = 2 · 35 · 2 + 3 · 21 · 1 + 2 · 15 · 1=233(5) 233 ≡ 23 (mod 105)
x≡ 2 (mod 3), x≡ 3 (mod 5), x≡ 2 (mod 7)
m 3 5 7 105
a 2 3 2
M 35 21 15
2.y1 1.y2 1.y3
y 2 1 1
2.35.2 3.21.1 2.15.1 233
233 ≡23) mod 105(
We conclude that 23 is the smallest positive integer that:
23 mod 3 = 223 mod 5 = 323 mod 7 = 2
Find all the solution to the system of congruencesx≡ 2 (mod 3), x≡ 1 (mod 4), x≡ 3 (mod 5)
x≡ 2 (mod 3), x≡ 1 (mod 4), x≡ 3 (mod 5)
m 3 4 5 60
a 2 1 3
M 20 15 12
2.y1 3.y2 2.y3
y 2 3 3
2.20.2 1.15.3 3.12.3 233
233≡ 53 (mod 60)
Home Work
Find all the solution to the system of congruencesx≡ 1 (mod 2), x≡ 2 (mod 3), x≡ 3 (mod 5),x≡ 4 (mod 11)
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