The Chinese Remainder Theorem (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 solution x = 0. The next question addresses these more general problems.
Prove that the solutions of the homogeneous version of the system of equations above (where ai= 0), are precisely the integer multiples of N= least common multiple (n1,..., nr).
Note: Compare this with the uniqueness statement in the CRT.
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 ChineseRemainderTheorem.
(ii) Find the 2 smallest positive integer solutions of the simultaneous set of congruence equations:
2x
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
Q1) Use the standard logical equivalences to simplify the expression
(ㄱp ^ q) v ㄱ(pVq)
Q2) consider the following theorem
"The square of every odd natural number is again an odd number"
What is the hypothesis of the theorem? what is the conclusion? give a direct proof of the theorem.
Q3) consider the follo
Find all solutions of each of the systems of congruences:-
(a) x is congruent to 1 (mod 2) (d) 4x is congruent to 2 (mod 6)
x is congruent to 2 (mod 3)
