Question

Solve the following linear congruence, 25x \equiv 15(\bmod 29)

Polynomial factorization
ANSWERED
asked 2021-05-09
Solve the following linear congruence,
\(\displaystyle{25}{x}\equiv{15}{\left({b}\text{mod}{29}\right)}\)

Answers (1)

2021-05-11
Step 1
The given linear congruence equation is
\(\displaystyle{25}{x}\equiv{15}{\left({b}\text{mod}{29}\right)}\)
Here, a=25, b=15, m=29,
And,
gcd(a,m)=gcd(25,29)=1
Hence the congruence has 1 incongruent solution which is given by solving the corresponding Diophantine equation
\(\displaystyle{a}{x}+{b}{y}={m}\Rightarrow{25}{x}+{15}{y}={29}\)
Step 2
Let us check the value of x=11.
That is, at x =11,
\(\displaystyle{11}\times{25}+{15}={275}+{15}={290}\)
which is \(\displaystyle{0}{b}\text{mod}{29}\).
Hence x=11 is the solution of given equation.
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