Solve the congruence x^2 -= 1 (mod 105)

Lipossig 2021-02-09 Answered
Solve the congruence x21(mod105)
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Expert Answer

Isma Jimenez
Answered 2021-02-10 Author has 84 answers
Step 1
Given:
x21(mod105)
To Solve:
The given congruence x21(mod105)
Step 2
For odd prime p and aZ, the Legendre symbol (ap) is given by,
(ap)={(0,ifpa,),(1,ifpdoes¬÷aandx2a(modp)hassolution.),(1,ifpdoes¬÷aandx21(modp)hasnosolution.)
Hence,(ap)=(bp)where,ab(modp)...(1)
The Legendre symbol can be evaluated by,
(ap)=ap12(modp)...(2)
Here,
The congruence is x21(mod105)
a=1 and p =105
The prime factorization of p is,
p=105=3×5×7
Thus,
(1105)=(13)(15)(17)
(13)=1312(mod3) (from using formula 2)
(13)=122(mod3)
(13)=1(mod3)
Now,
(15)=1512(mod5) (from using formula 2)
(15)=142(mod5)
(15)=1(mod5)
Now,
(17)=1712(mod7)
(17)=162(mod7)
(17)=1(mod7)
From above,

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