以下問題來自 Elementary Number Theory,Strayer此原文書裡面的習題問題!
請問高手能夠回答以下問題!
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或者回答其中幾題也是可以唷!感謝!
數論之The Chinese Remainder Theorem & Wilson's Theorem 9個問題(2008/05/12)
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 (m
1,m
2)|b
1-b
2.In this case,prove that the solution is unique modulo[m
1,m
2].
2.Prove that the system of linear congruences in one variable given by
•
•
•
is solvable if and only if if (m
1,m
2)|b
i-b
j for all i
j.In this case,prove that the solution is unique modulo [m
1,m
2,...,m
n].
3.
(a)Prove that if p is odd prime number,then
.
(b)Find the least nonnegative residue of 2(100!) modulo 103.
4.Let n
Z 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 p
1mod4,Prove that
is a solution of the quadratic congruence x
2-1mod p.
(c) If p
3mod4,Prove that
is a solution of the quadratic congruence x
21mod p.
8.Let p be an odd prime number.Prove that 1
23
25
2•••(p-4)
2(p-2)
2
9.Let p be a prime number congruent to 3 modulo 4.Prove that [(p-1)/2]!
.