Difference between revisions 644668518 and 685259365 on enwiki

{{Original research|date=September 2014}}
In [[modular arithmetic]], the '''method of successive substitution''' is a method of solving problems of [[simultaneous congruences]] by using the definition of the congruence equation. It is commonly applied in cases where the conditions of the [[Chinese remainder theorem]] are not satisfied.

(contracted; show full)====PROCEDURE====

1. Begin by rewriting the first congruence as an equation:
* '''x = 2a + 1, ∀a ∈ Z '''
2. Rewrite the second congruence as an equation, and set the equation found in the first step equal to this equation, since '''x '''will substitute the x in the second congruence:
* '''x ≡ 2 (mod 3)'''
* '''x = 2a + 1 ≡ 2 (mod 3)'''
* '''2a 
+≡ 1 = 3(mod 3)'''
* '''a +≡ 2   ''(Rewrite the second congruence in terms of its modulus)''⁻¹ (mod 3)'''
* '''a = -1.'''
Because '''a '''must be a [[Modular multiplicative inverse|positive nonnegative inverse]], we need a positive '''a'''. '''''Thus, we add whatever our current modulus is to a, which is a = -1 + 3 = 2'''''.

3. We now rewrite '''a = 2 '''in terms of our current modulus:
(contracted; show full)
==External links==
* [http://maciejkus.com/chinese_remainder/ A method of successive substitution calculator]

[[Category:Modular arithmetic]]
[[Category:Back Substitution]]
[[Category:Modular Arithmetic]]
[[Category:Modular Congruences]]