nyuuzyou/wikis huggingface dataset:
Divisor Count of 6561

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {6561} = 9$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$6561 = 3^8$
Thus:
The divisors of $6561$ can be enumerated as:
:$1, 3, 9, 27, 81, 243, 729, 2187, 6561$
Category:Examples of Divisor Count Function
Category:6561
Divisor Count of 672

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {672} = 24$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$672 = 2^5 \times 3 \times 7$
Thus:
The divisors of $672$ can be enumerated as:
:$1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 32, 42, 48, 56, 84, 96, 112, 168, 224, 336, 672$
Category:Examples of Divisor Count Function
Category:672
Divisor Count of 70

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {70} = 8$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$70 = 2 \times 5 \times 7$
Thus:
The divisors of $70$ can be enumerated as:
:$1, 2, 5, 7, 10, 14, 35, 70$
Category:Examples of Divisor Count Function
Category:70
Divisor Count of 714

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {714} = 16$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$714 = 2 \times 3 \times 7 \times 17$
Thus:
The divisors of $714$ can be enumerated as:
:$1, 2, 3, 6, 7, 14, 17, 21, 34, 42, 51, 102, 119, 238, 357, 714$
Category:Examples of Divisor Count Function
Category:714
Divisor Count of 720

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {720} = 24$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$720 = 2^4 \times 3^2 \times 5$
Thus:
The divisors of $720$ can be enumerated as:
:$1$, $2$, $3$, $4$, $5$, $6$, $8$, $9$, $10$, $12$, $15$, $16$, $18$, $20$, $24$, $30$, $36$, $40$, $45$, $48$, $60$, $72$, $80$, $90$, $120$, $144$, $180$, $240$, $360$, $720$
Category:Examples of Divisor Count Function
Category:720
Divisor Count of 720,720

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {720 \, 720} = 240$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$720 \, 720 = 2^4 \times 3^2 \times 5 \times 7 \times 11 \times 13$
Thus:
Category:Examples of Divisor Count Function
Category:720,720
Divisor Count of 748

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {748} = 12$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$748 = 2^2 \times 11 \times 17$
Thus:
The divisors of $748$ can be enumerated as:
:$1, 2, 4, 11, 17, 22, 34, 44, 68, 187, 374, 748$
Category:Examples of Divisor Count Function
Category:748
Divisor Count of 75

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {75} = 6$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$75 = 3 \times 5^2$
Thus:
The divisors of $75$ can be enumerated as:
:$1, 3, 5, 15, 25, 75$
Category:Examples of Divisor Count Function
Category:75
Divisor Count of 7560

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {7560} = 64$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$7560 = 2^3 \times 3^3 \times 5 \times 7$
Thus:
The divisors of $7560$ can be enumerated as:
:$1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 27, 28, 30, 35, 36, 40, 42,$
:$45, 54, 56, 60, 63, 70, 72, 84, 90, 105, 108, 120, 126, 135, 140, 168, 180, 189, 210,$
:$216, 252, 270, 280, 315, 360, 378, 420, 504, 540, 630, 756, 840, 945, 1080, 1260, 1512, 1890, 2520, 3780, 7560$
Category:Examples of Divisor Count Function
Category:7560
Divisor Count of 76

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {76} = 6$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$76 = 2^2 \times 19$
Thus:
The divisors of $76$ can be enumerated as:
:$1, 2, 4, 19, 38, 76$
Category:Examples of Divisor Count Function
Category:76
Divisor Count of 770

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {770} = 16$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$770 = 2 \times 5 \times 7 \times 11$
Thus:
The divisors of $770$ can be enumerated as:
:$1, 2, 5, 7, 10, 11, 14, 22, 35, 55, 70, 77, 110, 154, 385, 770$
Category:Examples of Divisor Count Function
Category:770
Divisor Count of 78

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {78} = 8$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$78 = 2 \times 3 \times 13$
Thus:
The divisors of $78$ can be enumerated as:
:$1, 2, 3, 6, 13, 26, 39, 78$
Category:Examples of Divisor Count Function
Category:78
Divisor Count of 8128

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {8128} = 10$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$8128 = 2^6 \times 127$
Thus:
The divisors of $496$ can be enumerated as:
:$1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064, 8128$
Category:Examples of Divisor Count Function
Category:8128
Divisor Count of 830,600

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {830 \, 600} = 24$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$830 \, 600 = 2^3 \times 5^2 \times 4153$
Thus:
Category:Examples of Divisor Count Function
Category:830,600
Divisor Count of 831,600

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {831 \, 600} = 240$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$831 \, 600 = 2^4 \times 3^3 \times 5^2 \times 7 \times 11$
Thus:
Category:Examples of Divisor Count Function
Category:831,600
Divisor Count of 836

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {836} = 12$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$836 = 2^2 \times 11 \times 19$
Thus:
The divisors of $836$ can be enumerated as:
:$1, 2, 4, 11, 19, 22, 38, 44, 76, 209, 418, 836$
Category:Examples of Divisor Count Function
Category:836
Divisor Count of 840

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {840} = 32$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$840 = 2^3 \times 3 \times 5 \times 7$
Thus:
The divisors of $840$ can be enumerated as:
:$1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, 840$
Category:Examples of Divisor Count Function
Category:840
Divisor Count of 88

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {88} = 8$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$88 = 2^3 \times 11$
Thus:
The divisors of $88$ can be enumerated as:
:$1, 2, 4, 8, 11, 22, 44, 88$
Category:Examples of Divisor Count Function
Category:88
Divisor Count of 90

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {90} = 12$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$90 = 2 \times 3^2 \times 5$
Thus:
The divisors of $90$ can be enumerated as:
:$1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90$
Category:Examples of Divisor Count Function
Category:90
Divisor Count of 910

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {910} = 16$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$910 = 2 \times 5 \times 7 \times 13$
Thus:
The divisors of $910$ can be enumerated as:
:$1, 2, 5, 7, 10, 13, 14, 26, 35, 65, 70, 91, 130, 182, 455, 910$
Category:Examples of Divisor Count Function
Category:910
Divisor Count of 9240

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {9240} = 64$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$9240 = 2^3 \times 3 \times 5 \times 7 \times 11$
Thus:
The divisors of $9240$ can be enumerated as:
:$1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 20, 21, 22, 24, 28, 30, 33, 35, 40, 42,$
:$44, 55, 56, 60, 66, 70, 77, 84, 88, 105, 110, 120, 132, 140, 154, 165, 168, 210, 220, 231,$
:$264, 280, 308, 330, 385, 420, 440, 462, 616, 660, 770, 840, 924, 1155, 1320, 1540, 1848, 2310, 3080, 4620, 9240$
== Sources ==
* : $9240$
* : $9240$
Category:Examples of Divisor Count Function
Category:9240
Divisor Count of 942,480

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {942 \, 480} = 240$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$942 \, 480 = 2^4 \times 3^2 \times 5 \times 7 \times 11 \times 17$
Thus:
Category:Examples of Divisor Count Function
Category:942,480
Divisor Count of 945

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {945} = 16$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$945 = 3^3 \times 5 \times 7$
Thus:
The divisors of $945$ can be enumerated as:
:$1, 3, 5, 7, 9, 15, 21, 27, 35, 45, 63, 105, 135, 189, 315, 945$
Category:Examples of Divisor Count Function
Category:945
Divisor Count of 98

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {98} = 6$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$98 = 2 \times 7^2$
Thus:
The divisors of $98$ can be enumerated as:
:$1, 2, 7, 14, 49, 98$
Category:Examples of Divisor Count Function
Category:98
Divisor Count of 982,800

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {982 \, 800} = 240$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$982 \, 800 = 2^4 \times 3^3 \times 5^2 \times 7 \times 13$
Thus:
Category:Examples of Divisor Count Function
Category:982,800
Divisor Count of 99

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {99} = 6$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$99 = 3^2 \times 11$
Thus:
The divisors of $99$ can be enumerated as:
:$1, 3, 9, 11, 33, 99$
Category:Examples of Divisor Count Function
Category:99
Divisor Count of 997,920

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {997 \, 920} = 240$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$997 \, 920 = 2^5 \times 3^4 \times 5 \times 7 \times 11$
Thus:
Category:Examples of Divisor Count Function
Category:997,920
Divisor Count of Square-Free Integer is Power of 2

== Theorem ==
Let $n$ be a square-free integer.
Let $\sigma_0: \Z \to \Z$ be the divisor count Function.
Then $\map {\sigma_0} n = 2^r$ for some $r \ge 1$.
The converse is not true in general.
That is, if $\map {\sigma_0} n = 2^r$ for some $r \ge 1$, it is not necessarily the case that $n$ is square-free.
== Proof ==
Let $n$ be an integer such that $n \ge 2$.
Let the prime decomposition of $n$ be:
:$n = p_1^{k_1} p_2^{k_2} \dotsm p_r^{k_r}$
Then from Divisor Count Function from Prime Decomposition we have that:
:$\ds \map {\sigma_0} n = \prod_{i \mathop = 1}^r \paren {k_i + 1}$
Let $n$ be square-free.
Then by definition:
:$\forall i: 1 \le i \le r: k_i = 1$
So:
:$\ds \map {\sigma_0} n = \prod_{i \mathop = 1}^r \paren {1 + 1} = 2^r$
The statement about the converse is proved by counterexample:
Let $n = p^3$ where $p$ is prime.
Then $n$ is not square-free as $p^2 \divides n$.
However:
:$\map {\sigma_0} n = 3 + 1 = 2^2$
Category:Divisor Count Function
Category:Square-Free Integers
Divisor Divides Multiple

== Theorem ==
Let $a, b$ be integers.
Let:
:$a \divides b$
where $\divides$ denotes divisibility.
Then:
:$\forall c \in \Z: a \divides b c$
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : Chapter $2$: Divisibility Theory in the Integers: $2.2$ The Greatest Common Divisor: Problems $2.2$: $2 \ \text{(a)}$
* : Chapter $2$: Integers and natural numbers: $\S 2.2$: Divisibility and factorization in $\mathbf Z$: $\mathbf D \, 5$
* : Appendix $1$: Some Results from the Elementary Theory of Numbers: Corollary $1$
Category:Divisors
Category:Divisor Divides Multiple
Divisor Divides Multiple/Proof 1

== Theorem ==
== Proof ==
Let $a \divides b$.
From Integer Divides Zero:
:$a \divides 0$
Thus $a$ is a common divisor of $b$ and $0$.
From Common Divisor Divides Integer Combination:
:$\forall p, q \in \Z: a \divides \paren {p \cdot b + q \cdot 0}$
Putting $p = c$ and $q = 1$ (for example):
:$a \divides \paren {c b + 0}$
Hence the result.
Category:Divisor Divides Multiple
Divisor Modulo Integer/Examples/8 modulo 12

== Examples of Divisors Modulo $m$ ==
In modulo $12$ division, $8$ has the following divisors:
:$1, 2, 4, 5, 7, 8, 10, 11$
== Proof ==
== Sources ==
* 
* 
Category:Examples of Divisors Modulo Integer
Divisor Relation in Integral Domain is Transitive

== Theorem ==
Let $\struct {D, +, \circ}$ be an integral domain.
Let $x, y, z \in D$.
Then:
:$x \divides y \land y \divides z \implies x \divides z$
=== Corollary ===
== Proof ==
Let $x \divides y \land y \divides z$.
Then from the definition of divisor, we have:
:$x \divides y \iff \exists s \in D: y = s \circ x$
:$y \divides z \iff \exists t \in D: z = t \circ y$
Then:
:$z = t \circ \paren {s \circ x} = \paren {t \circ s} \circ x$
Thus:
:$\exists \paren {t \circ s} \in D: z = \paren {t \circ s} \circ x$
and the result follows.
== Sources ==
* : Chapter $6$: Polynomials and Euclidean Rings: $\S 26$. Divisibility: Theorem $49 \ \text{(i)}$
Category:Integral Domains
Divisor Relation induces Lattice

== Theorem ==
Let $\struct {\Z_{> 0}, \divides}$ be the ordered set comprising:
:The set of positive integers $\Z_{> 0}$
:The divisor relation $\divides$ defined as:
::$a \divides b := \exists k \in \Z_{> 0}: b = ka$
Then $\struct {\Z_{> 0}, \divides}$ is a lattice.
== Proof ==
It follows from Divisor Relation on Positive Integers is Partial Ordering that $\struct {\Z_{> 0}, \divides}$ is indeed an ordered set.
Let $a, b \in \Z_{>0}$.
Let $d = \gcd \set {a, b}$ be the greatest common divisor of $a$ and $b$.
By definition, $d$ is the infimum of $\set {a, b}$.
Similarly, let $m = \lcm \set {a, b}$ be the lowest common multiple of $a$ and $b$.
By definition, $m$ is the supremum of $\set {a, b}$.
Hence, as $\set {a, b}$ has both an infimum and a supremum in $\Z_{>0}$, it follows that $\struct {\Z_{> 0}, \divides}$ is a lattice.
== Sources ==
* : $\S 7$: Example $7.8$
Category:Lattice Theory
Category:Number Theory
Divisor Relation is Antisymmetric

== Theorem ==
Divides is a antisymmetric relation on $\Z_{>0}$, the set of positive integers.
That is:
:$\forall a, b \in \Z_{>0}: a \divides b \land b \divides a \implies a = b$
=== Corollary ===
== Proof ==
Let $a, b \in \Z_{> 0}$ such that $a \divides b$ and $b \divides a$.
Then:
If we restrict ourselves to the domain of positive integers, we can see:
:$\forall a, b \in \Z_{>0}: a \divides b \land b \divides a \implies a = b$
Category:Divisors
Category:Divisor Relation is Antisymmetric
Divisor Relation is Antisymmetric/Corollary

== Corollary to Divisor Relation is Antisymmetric ==
Let $a, b \in \Z$.
If $a \divides b$ and $b \divides a$ then $a = b$ or $a = -b$.
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : Chapter $2$: Divisibility Theory in the Integers: $2.2$ The Greatest Common Divisor: Theorem $2 \text{-} 2 \ (5)$
* : Chapter $2$: Integers and natural numbers: $\S 2.2$: Divisibility and factorization in $\mathbf Z$: $\mathbf D. \, 3$
Category:Divisor Relation is Antisymmetric
Divisor Relation is Antisymmetric/Corollary/Proof 1

== Corollary to Divisor Relation is Antisymmetric ==
== Proof ==
Let $a \divides b$ and $b \divides a$.
Then from Divisor Relation is Antisymmetric:
:$\size a = \size b$
The result follows from Integer Divides its Negative and Integer Divides its Absolute Value.
Category:Divisor Relation is Antisymmetric
Divisor Relation is Antisymmetric/Corollary/Proof 2

== Corollary to Divisor Relation is Antisymmetric ==
== Proof ==
Let $a \divides b$ and $b \divides a$.
Then by definition of divisor:
:$\exists c, d \in \Z: a c = b, b d = a$
Thus:
== Sources ==
* : Introduction $\S 6$: The division process in $I$
* : Chapter $3$: The Integers: $\S 10$. Divisibility: Theorem $17 \ \text{(ii)}$
* : Chapter $2$: Integers and natural numbers: $\S 2.2$: Divisibility and factorization in $\mathbf Z$
Category:Divisor Relation is Antisymmetric