## [數學]數論之The Chinese Remainder Theorem & Wilson's Theorem 9個問題5/12

### [數學]數論之The Chinese Remainder Theorem & Wilson's Theorem 9個問題5/12

1~2題為The Chinese Remainder Theorem,3~9題為Wilson's Theorem
1.Prove that the system of linear congruent in one variable given by

is solvable if and only if (m1,m2)|b1-b2.In this case,prove that the solution is unique modulo[m1,m2].
2.Prove that the system of linear congruences in one variable given by

is solvable if and only if  if (m1,m2)|bi-bj for all ij.In this case,prove that the solution is unique modulo [m1,m2,...,mn].
3.
(a)Prove that if p is odd prime number,then.
(b)Find the least nonnegative residue of 2(100!) modulo 103.
4.Let nZ with n>1.Prove that n is a prime number if and only if (n-2)!1 mod n.
5.Let n be a composite integer greater than 4.Prove that (n-1)!0mod n.
6.Let p be a prime number.Prove that the numerator of 1+ is divisible by p.
7.Let p be an odd prime number.
(a)Prove that .
(b)If p1mod4,Prove that is a solution of the quadratic congruence x2-1mod p.
(c) If p3mod4,Prove that is a solution of the quadratic congruence x21mod p.
8.Let p be an odd prime number.Prove that 123252•••(p-4)2(p-2)2
9.Let p be a prime number congruent to 3 modulo 4.Prove that [(p-1)/2]!.

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