Find all solutions to the following system of linear congruences: x-= 1 mod 2, x -= 2 mod 3, x-= 3 mod 5, x -= 4 mod 7, x -= 5 mod 11.

Ramsey 2021-02-04 Answered
Find all solutions to the following system of linear congruences: x1mod2,x2mod3,x3mod5,x4mod7,x5mod11.
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Expert Answer

oppturf
Answered 2021-02-05 Author has 94 answers

Step 1
The system of linear congruence’s are given by,
x1mod2
x2mod3
x3mod5
x4mod7
x5mod11
Step 2
According to the Chinese Remainder Theorem,
M=235711=2310
M1=M2=23102=1155
M2=M3=23103=770
M3=M5=23105=462
M4=M7=23107=330
M5=M11=231011=210
Step 3
Find the value of y1,y2,y3,y4,y5.
1y11mod2y1=1
2y21mod3y2=2
2y31mod5y3=3
1y41mod7y4=1
1y51mod11y5=1
Step 4
Solution is,
x=(1M1y1+2M2y2+3M3y3+4M4y4+5M5y5)modM
x=(1×1155×1+2×770×2+3×462×3+4×330×1+5×210×1)mod2310
x=10763mod2310
x=1403mod2310

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