number theory congruence problems and solutions


What is congruence in number theory?

    Some texts on number theory use this latter relationship as the definition of congruence: Two integers a and b are said to be congruent modulo n, if n | ( a b ). Using this latter definition as the starting point, prove that if ( a mod n) = ( b mod n ), then n divides ( a b ).

What is the proof of congruence?

    The proof begins by justifying that ¯¯¯¯¯¯¯¯AC ?¯¯¯¯¯¯¯¯CB A C ¯ ? C B ¯, which shows that those two sides are congruent. Also, we know from the given statement that the other two corresponding pairs of sides are congruent.

Why is it difficult to choose a congruence test?

    When proving results involving similarity and congruence, some students may still find it challenging to decide which test to use. Problems involving equality of lengths usually involve congruence. Problems involving proportions involve similarity. In selecting a congruence test, students may make these errors:

What are the basic properties of a congruence relation?

    Basic properties[edit] The congruence relation satisfies all the conditions of an equivalence relation: Reflexivity: a? a(mod n) Symmetry: a? b(mod n)if b? a(mod n). Transitivity: If a? b(mod n)and b? c(mod n), then a? c(mod n) If a1? b1(mod n)and a2? b2(mod n),or if a? b(mod n),then:
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