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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"?