4383_Notes_4o4_fill - When b = 1 the equivalence class is...

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Number Theory Chapter 4 Linear Congruence and the Chinese Remainder Theorem Linear Congruence Given integers a and b and n>1, find a solution to the equation ( mod ax b n . What this means: Congruent solutions versus incongruent solutions: Solutions in terms of the equivalence classes mod n: We can think of this equation as one in the arithmetic of n find an integer x such that the equation [a] * [x] = [b]
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The relationship between this and linear Diophantine equations When is there a solution?
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If gcd(a, n) = 1, the linear equation ( mod ax b n has a unique solution modulo n. Or, there exists a unique element [x] such that [a] * [x] = [b]
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Unformatted text preview: When b = 1 the equivalence class is called the multiplicative inverse in ℤ Examples: What about SYSTEMS of linear congruence equations? ( 29 ( 29 ( 29 1 1 1 2 2 2 mod mod ... mod k a x b m ≡ ≡ ≡ When does it have a solution? Why can we reduce this to a system where all the moduli are relatively prime? Early Chinese contributions to this type of problem occur in the literature as early as the first century AD – so the method of solution is called the Chinese Remainder Theorem. The Chinese Remainder Theorem Example Example:...
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