Given the following system of linear simultaneous congruences what is the value of x that satisfies them all? x -= 18 mod 29 x -= 20 mod 31 x -= 6 mod 17

Dottie Parra 2020-11-12 Answered
Given the following system of linear simultaneous congruences what is the value of x that satisfies them all?
x18mod29
x20mod31
x6mod17
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Expert Answer

davonliefI
Answered 2020-11-13 Author has 79 answers
Step 1
According to the given information, it is required to find the value of x that satisfies all simultaneous congruence.
x18mod29
x20mod31
x6mod17
Step 2
Using Chinese remainder theorem:
Let m1,m2,.mr be a collection of pairwise relative prime integers.
then the system of simultaneous congruence
xa1(modm1)
xa2(modm2)
....
x=ar(modmr)
has a unique solution modulo M=m1m2mr for any integers a1,a2,ar
Step 3
Now using the above theorem solve the given question.
a1=18,a2=20,a3=6
M=29×31×17=15283
m1=1528329=527
m2=1528331=493
m3=1528317=899
Step 4
Solving further:
m1m1=1(mod29)527m1=1(mod29)
5m1=1(mod29)m1=35
m2m2=1(mod31)493m2=1(mod31)
3m2=1(mod31)m2=10
m3m3=1(mod17)899m3=1(mod17)
15m3=1(mod17)m3=8
Step 5
Therefore, the value of x is:
x=a1m1m1+a2m2m2+a3m3m3
=(18×527×35)+(20×493×10)+(6×899×8)
=332010+98600+43152
=473762(mod15283)
x=15272
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