I have difficulty understanding Modulo ? What is its meaning?

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I have difficulty understanding "Modulo"? What is its meaning?

[tex]a\equiv b\left(mod n\right)[/tex]
means a is "congruent" to b "modulo" n
means a-b=kn


Except accepting this is an equivalence relation, I feel very uncomfortable about "modulo" because of the lack of visual picture of this concept.
Everytime i have to move the "b" to the left side and see if "k" exists. This is very annoying and it totally shuts down my math intuition.
Why do mathematicians introduce this concept? Will it lead to any profound meaning/result? Also as "a" and "b" can be very far away, why use the word "congruent"?
 

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HallsofIvy
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[tex]a\equiv b\left(mod n\right)[/tex]
means a is "congruent" to b "modulo" n
means a-b=kn


Except accepting this is an equivalence relation, I feel very uncomfortable about "modulo" because of the lack of visual picture of this concept.
Everytime i have to move the "b" to the left side and see if "k" exists. This is very annoying and it totally shuts down my math intuition.
Why do mathematicians introduce this concept? Will it lead to any profound meaning/result? Also as "a" and "b" can be very far away, why use the word "congruent"?
"congruent" has nothing to do with be close. "Congruent" means "the same in this particular way" where "this particular way" is defined for that particular use of "congruent".

In "modulo" arithmetic [itex]a\equiv b (mod n)[/itex] if and only if dividing each by n gives the same remainder. [itex]7\equive 19 (mod 4)[/itex] because dividing 7 by 4 gives a quotient of 1 and a remainder of 3 while dividing 19 by 4 gives a quotient of 4 and a remainder of 3. We are ignoring the quotient and looking only at the remainder.
 
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"congruent" has nothing to do with be close. "Congruent" means "the same in this particular way" where "this particular way" is defined for that particular use of "congruent".

In "modulo" arithmetic [itex]a\equiv b (mod n)[/math] if and only if dividing each by n gives the same remainder. [itex]7\equive 19 (mod 4)[/math] because dividng 7 by 4 gives a quotient of 1 and a remainder of 3 while dividing 19 by 4 gives a quotient of 4 and a remainder of 3. We are ignoring the quotient and looking only at the remainder.

thanks hallsofivy. excellent examples and explanations.
 

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