Add divisibility, primes, congruence and factorial

_number-systems.qmd

@@ -7,6 +7,8 @@ and measure with the real numbers.
 This page defines those number systems.
 It also defines the words that sort their members:
 even and odd integers, and rational and irrational real numbers.
+Finally, it defines divisibility, [primes](#def-prime) and congruence,
+and the [factorial](#def-factorial) of a [non-negative integer](#def-nonnegative-integers).
 
 :::
 
@@ -50,6 +52,38 @@ even and odd integers, and rational and irrational real numbers.
 
 {{< include mds/_subfiles/number-systems/_note-natural-numbers-other-conventions.qmd >}}
 
+{{< slidebreak >}}
+
+{{< include mds/_subfiles/number-systems/_def-factorial.qmd >}}
+
+{{< slidebreak >}}
+
+{{< include mds/_subfiles/number-systems/_exm-factorial.qmd >}}
+
+# Divisibility, primes and congruence {#sec-number-systems-divisibility}
+
+{{< include mds/_subfiles/number-systems/_def-divides.qmd >}}
+
+{{< slidebreak >}}
+
+{{< include mds/_subfiles/number-systems/_exm-divides.qmd >}}
+
+{{< slidebreak >}}
+
+{{< include mds/_subfiles/number-systems/_def-prime.qmd >}}
+
+{{< slidebreak >}}
+
+{{< include mds/_subfiles/number-systems/_exm-prime.qmd >}}
+
+{{< slidebreak >}}
+
+{{< include mds/_subfiles/number-systems/_def-congruence.qmd >}}
+
+{{< slidebreak >}}
+
+{{< include mds/_subfiles/number-systems/_exm-congruence.qmd >}}
+
 # Rational, real, and irrational numbers {#sec-notation-reals}
 
 {{< include mds/_subfiles/number-systems/_def-rational-numbers.qmd >}}

_subfiles/number-systems/_def-congruence.qmd

@@ -0,0 +1,13 @@
+:::{#def-congruence}
+#### Congruence modulo $n$
+
+Let $a$ and $b$ be [integers](#def-integers),
+and let $n$ be a [positive integer](#def-positive-integer).
+We say that $a$ is **congruent to $b$ modulo $n$**,
+written $a \equiv b \pmod{n}$,
+if $n$ [divides](#def-divides) $a - b$, written $n \mid (a - b)$,
+[@wikipedia_modular_arithmetic].
+When $a$ is not congruent to $b$ modulo $n$,
+we write $a \not\equiv b \pmod{n}$.
+
+:::

_subfiles/number-systems/_def-divides.qmd

@@ -0,0 +1,11 @@
+:::{#def-divides}
+#### Divides
+
+Let $a$ and $b$ each be an [integer](#def-integers).
+We say that $a$ **divides** $b$,
+written $a \mid b$,
+if $b = a c$ for some integer $c$.
+When $a$ does not divide $b$,
+we write $a \nmid b$.
+
+:::

_subfiles/number-systems/_def-factorial.qmd

@@ -0,0 +1,12 @@
+:::{#def-factorial}
+#### Factorial
+
+For a [non-negative integer](#def-nonnegative-integers) $n$,
+the **factorial** of $n$ [@wikipedia_factorial],
+written $\factorial{n}$,
+is $1 \cdot 2 \cdots n$,
+the [natural numbers](#def-natural-numbers) from 1 to $n$ multiplied together.
+For $n = 0$ there is nothing to multiply,
+and $\factorial{0} = 1$.
+
+:::

_subfiles/number-systems/_def-prime.qmd

@@ -0,0 +1,8 @@
+:::{#def-prime}
+#### Prime number
+
+An [integer](#def-integers) $n \ge 2$ is **prime**
+if no [positive integer](#def-positive-integer) other than $1$ and $n$ [divides](#def-divides) it
+[@wikipedia_prime_number].
+
+:::

_subfiles/number-systems/_exm-congruence.qmd

@@ -0,0 +1,15 @@
+:::{#exm-congruence}
+#### Clock arithmetic
+
+- $17$ is [congruent to $5$ modulo $12$](#def-congruence), $17 \equiv 5 \pmod{12}$,
+  because $17 - 5 = 12$
+  and $12 \mid 12$.
+  On a 12-hour clock,
+  hour 17 is 5 o'clock.
+- $17 \not\equiv 4 \pmod{12}$,
+  because $17 - 4 = 13$
+  and $12 \nmid 13$.
+
+<!-- no-figure: two single congruences -->
+
+:::

_subfiles/number-systems/_exm-divides.qmd

@@ -0,0 +1,15 @@
+:::{#exm-divides}
+#### Divisors and non-divisors
+
+- $3$ [divides](#def-divides) $12$, $3 \mid 12$,
+  because $12 = 3 \cdot 4$
+  and $4$ is an [integer](#def-integers).
+- $5 \nmid 12$,
+  because $5 \cdot 2 = 10 < 12 < 15 = 5 \cdot 3$,
+  so no integer $c$ has $5 c = 12$.
+- Every integer $a$ [divides](#def-divides) $0$,
+  because $0 = a \cdot 0$.
+
+<!-- no-figure: three single divisibility facts -->
+
+:::

_subfiles/number-systems/_exm-factorial.qmd

@@ -0,0 +1,10 @@
+:::{#exm-factorial}
+#### Factorials
+
+- $\factorial{3} = 1 \cdot 2 \cdot 3 = 6$.
+- $\factorial{4} = 1 \cdot 2 \cdot 3 \cdot 4 = 24$.
+- $\factorial{0} = 1$, by the [definition of the factorial](#def-factorial).
+
+<!-- no-figure: three single values -->
+
+:::

_subfiles/number-systems/_exm-prime.qmd

@@ -0,0 +1,15 @@
+:::{#exm-prime}
+#### Primes and non-primes
+
+- $7$ is [prime](#def-prime):
+  none of $2$, $3$, $4$, $5$ and $6$ [divides](#def-divides) $7$,
+  and no [integer](#def-integers) above $7$ does,
+  so its only positive divisors are $1$ and $7$.
+- $9$ is not prime,
+  because $3$ divides $9$.
+- $1$ is not prime,
+  because the definition requires $n \ge 2$.
+
+<!-- no-figure: three single numbers -->
+
+:::

references.bib

@@ -2846,3 +2846,27 @@ @inproceedings{loshchilov2017sgdr
   doi       = {10.48550/arXiv.1608.03983},
   url       = {https://arxiv.org/abs/1608.03983}
 }
+
+@misc{wikipedia_prime_number,
+  author = {{Wikipedia contributors}},
+  title = {Prime number --- {Wikipedia}{,} The Free Encyclopedia},
+  year = {2026},
+  url = {https://en.wikipedia.org/wiki/Prime_number},
+  note = {[Online; accessed 23-October-2026]}
+}
+
+@misc{wikipedia_modular_arithmetic,
+  author = {{Wikipedia contributors}},
+  title = {Modular arithmetic --- {Wikipedia}{,} The Free Encyclopedia},
+  year = {2026},
+  url = {https://en.wikipedia.org/wiki/Modular_arithmetic},
+  note = {[Online; accessed 23-October-2026]}
+}
+
+@misc{wikipedia_factorial,
+  author = {{Wikipedia contributors}},
+  title = {Factorial --- {Wikipedia}{,} The Free Encyclopedia},
+  year = {2026},
+  url = {https://en.wikipedia.org/wiki/Factorial},
+  note = {[Online; accessed 24-October-2026]}
+}