Congratulations, you have reached the end of this lesson! Let’s recap what we learned:

• We say that, for integers a, b, and n with n > 0, a is congruent to b mod n if n divides (a-b). Equivalently, a is congruent to b mod n if there exists some integer k for which a-b = n*k is true. We write this mathematically as:

$a \equiv b\;(mod\;n)$

• The reflexive property for congruences states that a is congruent to a mod n.

• The symmetric property for congruences states that, if a is congruent to b mod n, then b is congruent to a mod n.

• The transitive property for congruences says that, if a is congruent to b mod n and b is congruent to c mod n, then a is congruent to c mod n

• We can solve for congruences and linear congruences using a procedure that is analogous to solving for the variable x in a linear equation. The only differences are that congruences must perform modular arithmetic after each step and that the solution of a congruence problem has many possibilities for the value of the unknown variable.