# Wikipedia Authors - Modular arithmetic (Highlights)  ## Metadata **Review**:: [readwise.io](https://readwise.io/bookreview/38412916) **Source**:: #from/readwise #from/reader **Zettel**:: #zettel/fleeting **Status**:: #x **Authors**:: [[Wikipedia Authors]] **Full Title**:: Modular arithmetic **Category**:: #articles #readwise/articles **Category Icon**:: 📰 **URL**:: [en.wikipedia.org](https://en.wikipedia.org/wiki/Modular_arithmetic#Integers_modulo_n) **Host**:: [[en.wikipedia.org]] **Highlighted**:: [[2024-03-06]] **Created**:: [[2024-03-06]] ## Highlights - Given an [integer](https://en.wikipedia.org/wiki/Integer) *m* ≥ 1, called a **modulus**, two integers a and b are said to be **congruent** modulo m, if m is a [divisor](https://en.wikipedia.org/wiki/Divisor) of their difference ([View Highlight](https://read.readwise.io/read/01hr8z16x8cz0mga3h5gsc48bx)) ^688724543 #definition - Rather, *a* ≡ *b* (mod *m*) asserts that *a* and *b* have the same remainder when divided by *m*. ([View Highlight](https://read.readwise.io/read/01hr8z0wkbs12znhsbn53qrqbg)) ^688724466 <!-- New highlights added March 30, 2024 at 10:14 AM --> - The ring of integers modulo *m* is a [field](https://en.wikipedia.org/wiki/Field_(mathematics)) if and only if *m* is [prime](https://en.wikipedia.org/wiki/Prime_number) (this ensures that every nonzero element has a [multiplicative inverse](https://en.wikipedia.org/wiki/Modular_multiplicative_inverse)). If $m = p^k$ is a [prime power](https://en.wikipedia.org/wiki/Prime_power) with *k* > 1, there exists a unique (up to isomorphism) finite field $\mathrm{GF}(m) = \mathbb{F}_m$ with *m* elements, which is *not* isomorphic to $\mathbb{Z}/m\mathbb{Z}$, which fails to be a field because it has [zero-divisors](https://en.wikipedia.org/wiki/Zero-divisor). ([View Highlight](https://read.readwise.io/read/01ht6hw38cvebfp5mj53kbdrk7)) ^699857992