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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:
ab(modn)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.

Instructions

1.

Suppose we have the following congruence:

5a8(mod5)5a \equiv 8\;(mod\;5)

Create a variable called answer and set it to either True if you think this congruence can be solved or False if you think this congruence cannot be solved.

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