# Which role does modular arithmetic play in number theory?

I try to get a feeling for the role modular arithmetic does play in (elementary) number theory.

In the Wikipedia article on number theory, modular arithmetic isn't mentioned explicitly, only modular forms. But it is listed as a key concept (but not an advanced one). And there is this footnote:

See the comment on the importance of modularity in Iwaniec & Kowalski 2004, p. 1

Modular arithmetic might be described as the study of divisibility relations between integer numbers in the form of equations. I assume this is a main part of elementary number theory. But if it's not all that is elementary number theory about, what else?

Finally: Where outside of number theory does modular arithmetic play a role? Or only as a part of number theory? Possibly in a field like arithmetic geometry?

• There is a link at the end of the article to another article "Modular Arithmetic". Also, while telling about Gauss it states "He also introduced some basic notation (congruences)" which is the key feature of modular arithmetic. – Somos Dec 12 '18 at 15:41