Methodsuccessive substitution

The methodsuccessive substitution,mathematics, ismethodsolving problemssimultaneous congruences by usingdefinition ofcongruence equation.

For example, considersimple setsimultaneous congruences

x ≡ 3 (mod 4)

x ≡ 11 (mod 12)

Now,x ≡ 3 (mod 4)be true, x=3+4jsome integer j. Substitute this insecond equation

3+4j ≡ 11 (mod 12)

since welooking forsolutionboth equations.

Subtract 3 from both sides (thispermittedmodular arithmetic)

4j ≡ 11 (mod 12)

We needfindmultiplicative inverse4 mod 12, which we can find as 4. Multiply throughoutget

j ≡ 44 (mod 12)

j ≡ 8 (mod 12)

Forabovebe true, j=8+12ksome integer k. Now substitute back into 3+4jwe obtain

x=3+4(8+12k)

Expand out

x=35+48k

to obtainsolution

x≡ 35 (mod 48)

In general:

• writefirst equationits equivalent form
• substituteintonext
  • simplify, usemultiplicative inverse if necessary
• continue untillast equation
• back substitute, then simplify
• rewrite back incongruence form