I would appreciate it if someone could provide the solutions to QB5 of the attatched exam paper.
Please see the attached file for the fully formatted problems.

B5.
(a) (1) State and prove the Chinese Remainder Theorem.
(ii) Find the 2 smallest positive integer solutions of the simultaneous set of congruence equations:
2x=3 (mod 5)
3x=4 (mod 7)
x=5 (mod8)
(b) Let p be a prime and a a positive integer. How many solutions are there to the equation x2 ? x O(mod pr')?
(c) Let n and in be coprirne integers. Show ? x 0 (mod nrn) if and only if x2 ? x 0 (mod n) and ? x 0 (mod m).
(d) How many solutions are there to the equation x2 ? x 0 (mod N)
where N has collected prime factorization N = .

Please see the attached file for the full solution.
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QB5:
a. Proof:
1. Chinese Remainder Theorem: Suppose are positive integers relatively prime to each other, then for any integers with , must have a unique solution. This solution is

where and ,
Proof: Since are relatively ...

Solution Summary

The Chinese Remainder Theorem is Proven and Problems are solved. The solution is detailed and well presented. The solution was given a rating of "5" by the student who originally posted the question.

The ChineseRemainderTheorem (CRT) applies when the moduli ni in the system of equations x≡ a1 (mod n1) ... x≡ ar (mod nr) are pairwise relatively prime. When they are not, solutions x may or may not exist. However, the related homogeneous system (2'), in which all ai=0, always has a solution, namely the trivial

Solve the egg problem used to illustrate the ChineseRemainderTheorem if the remainders on division by 2,3,4,5,6 and 7 are all 1 by
(i) the given procedure (ii) the "easier" way
(i) the given procedure
X=1(Mod2)
X=1(Mod3)
X=1(Mod4)
X=1(Mod5)
X=1(Mod6)
X=1(Mod7)
If andand then . We will use this r

I need help to writing a program "VC++.net"with the specified input and output.
Please, Implement the ChineseRemainderTheorem. Allowing at least 3 pairwise relatively prime positive integers.
Please attention
This program must run in the Visual C++.NET. Some time I have problem with that, so, please the whole project in

I need help to writing a program "VC++.net"with the specified input and output.
Please, Implement the ChineseRemainderTheorem. Allowing at least 3 pairwise relatively prime positive integers.
Example:
Problem #1
Solve
p1: x = 2 (mod 3)
p2: x = 3 (mod 5)
p3: x = 2 (mod 7)
From p1, x = 3t + 2, for some in

1) Derive a formula to find all numbers that meet the following conditions:
• Divide by 3 the remainder is 1
• Divide by 29 the remainder is 6
2) What are the first 20 solutions to Sun Tsu's ChineseRemainder problem?
3) Derive a formula to find all numbers that meet the following conditions:
• Divide by 3 the r

1. Describe the history of the ChineseRemainderTheorem. Describe some of the relevant problems posed in Chineseand Hindu writings and how the ChineseRemainderTheorem applies to them. Please show references.

1. Without assuming Theorem 2-1, prove that for each pair of integers j and k (k > 0), there exists some integer q for which j ? qk is positive.
2. The principle of mathematical induction is equivalent to the following statement, called the least-integer principle:
Every non-empty set of positive integers has a least element.