Solve the system of linear congruences: 7x = 1 mod 8, 7x = 1 mod 13, 7x = 1 mod 17
Solve the system of linear congruences: 7x = 1 mod 8, 7x = 1 mod 13, 7x = 1 mod 17
Homework Answers
7x=1 mod 8
-x=1 mod 8
x=-1=1 mod 8
x=8k+1
7(8k+1)=1 mod 13
56k=-6 mod 13
28k=-3 mod 13
2k=-3 mod 13
Multiplying by 6 gives
12k=-6 mod 13
-k=-6 mod 13
k=6 mod 13
k=13p+6
x=8k+1=104p+49
7x=1 mod 17
We first convert coefficinet of x to 1
5*7x=5 mod 17
x=5 mod 17
104p+49=5 mod 17
2p=-44 mod 17
2p=-51+7=7 mod 17
2p=7 mod 17
Multiplying by 9
18p=p=63=12 mod 17
p=12 mod 17
p=17q+12
x=104p+49=104*17q+104*12+49
x=1768q+1297
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Solve the system of linear congruences: 7x = 1 mod 8, 7x = 1 mod 13, 7x = 1 mod 17
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