Solution of Linear Congruence - congruences | linear congruence and its solution | solve the linear congruence 12x≡9(mod 15)
A congruence of the form ax≡b(mod m) where x is an unknown integer is called a linear congruence in one variable.
It is important to know that if x0 is a solution for a linear congruence, then all integers xi such that xi≡x0(mod m) are solutions of the linear congruence. Notice also that ax≡b(mod m) is equivalent to a linear Diophantine equation i.e. there exists y such that ax−my=b . We now prove theorems about the solutions of linear congruences.
Let a,b and m be integers such that m is positive and let c=(a,m) . If c does not divide b , then the congruence ax≡b(mod m) has no solutions. If c∣b , then
ax≡b(mod m)
has exactly c incongruent solutions modulo m .
As we mentioned earlier, ax≡b(mod m) is equivalent to ax−my=b . By Theorem 19 on Diophantine equations, we know that if c does not divide b , then the equation, ax−my=b has no solutions. Notice also that if c∣b , then there are infinitely many solutions whose variable x is given by
x=x0+(m/c)t
Thus the above values of x are solutions of the congruence ax≡b(mod m) . Now we have to determine the number of incongruent solutions that we have. Suppose that two solutions are congruent, i.e.
x0+(m/c)t1≡x0+(m/c)t2(mod m).
Thus we get
(m/c)t1≡(m/c)t2(mod m).
Now notice that (m,m/c)=m/c and thus
t1≡t2(mod c).
Thus we get a set of incongruent solutions given by x=x0+(m/c)t , where t is taken modulo c .
Notice that if c=(a,m)=1 , then there is a unique solution modulo m for the equation ax≡b(mod m) .
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