A linear congruence equation is equivalent to a linear equation where all coefficients and all variables are from the Set of Integers (Z). The linear congruence equation ax = b (mod n) may be rewritten as ax1 = b - nx2 where x1, x2 -E- Z. For example 25x = 15 (mod 29) may be rewritten as 25x1 = 15 - 29x2.

$\begingroup$ @Gandhi, the first two parts are standard results about linear congruences, discussed in many intro Number Theory textbooks.

Rather, this is linear algebra. Instead, you are working with a $2\times2$ linear system over a given modulus, $9$. Here, the first two elementary methods of solving linear systems apply: substitution and elimination. The difference, however, is that we cannot generally divide by anything sharing divisors with $9$, i.e. multiples of $3$.

Jul 24, 2012 · When I am faced with a simple linear congruence such as $$9x \equiv 7 \pmod{13}$$ and I am working without any calculating aid handy, I tend to do something like the following: "Notice" that adding $13$ on the right and subtracting $13x$ on the left gives: $$-4x \equiv 20 \pmod{13}$$

Dividing the linear congruence equations. Ask Question Asked 11 years, 8 months ago.

Mar 4, 2013 · I'm trying to solve a linear congruence and I don't understand how some of them are reduced. For example, How does $105k \equiv -32 \pmod{11}\,$ reduce to $6k \equiv 1 \pmod{11}$?

Solving non-linear congruence. Ask Question Asked 12 years, 2 months ago. Modified 12 years, 2 months ago.

Jan 12, 2019 · Solving a Linear Congruence. Related. 1. Congruence Relation with exponents and variables. 1.

Aug 2, 2018 · The incongruent solutions of a linear congruence. 1. Number of solutions of linear congruence. 2.

Jan 5, 2019 · My question is to do with the incongruent solutions of a linear congruence. This is the problem: Find all integer solutions to the linear congruence $15x \equiv 36 \mod 57$ .