Dǣlere
In [[rīmcræft]]um, an '''dǣlere''' for þæt [[tæl]] ''n'' is tæl þæt dǣlþ ''n'' unoferēacalīċe.  Tō bisene, 7 is dǣlere for 42 for þǣm 42/7 = 6.  Man ēac sǣȝþ þæt "42 is '''tōdǣlliċ''' be 7" oþþe "42 is '''mārafeald''' of 7" oþþe "7 '''dǣlþ''' 42" and man oft ƿrītþ  7 | 42.

An '''dǣlere''' is ēac '''lǣssadǣl''' of ''n'' ȝehāten.

Dǣleras cunnon positife oþþe negatife bēon.  Þā positife dǣleras for 42 sind {1, 2, 3, 6, 7, 14, 21, 42}.

Sum synderlīpe talu: 1 and -1 sind dǣleras for ealle talu, and ȝehƿilċ tæl is dǣlere of 0.  Þā talu þæt 2 dǣlþ ȝehāten [[efentæl and ofertæl|efentalu]] and þā oðrum talum ȝehāten [[efentæl and ofertæl|ofertalu]].

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== Regolas for þǣm lytlum dǣlerum ==

Sind sume regolas þe ālīefþ þæt man oncnƿǣþ smale dǣleras sumes tæles be  allow to recognize small divisors of a number from the number's decimal digits:


A [[divisibility rule]] is a rule you can use to determine a number's divisibility by another number. In [[decimal]], the divisibility rules are:

* sum getæl is dǣlendlic þurh [[2 (number)|2]] [[iff]] se endemesta rīm is tōdǣlendic þurh 2
* sum getæl is dǣlendlic þurh [[3 (number)|3]] iff the sum of its digits is divisible by 3
* sum getæl is dǣlendlic þurh [[4 (number)|4]] iff the number given by the last two digits is divisible by 4
* sum getæl is dǣlendlic þurh [[5 (number)|5]] iff the last digit is [[zero|0]] or 5
* sum getæl is dǣlendlic þurh [[6 (number)|6]] iff it is divisible by 2 and by 3
* sum getæl is dǣlendlic þurh [[7 (number)|7]] iff the result of subtracting twice the last digit from the number with the last digit removed is divisible by 7 (e.g. 364 is divisible by 7 since 36-2&times;4 = 28 is divisible by 7)
* sum getæl is dǣlendlic þurh [[8 (number)|8]] iff the number given by the last three digits is divisible by 8
* sum getæl is dǣlendlic þurh [[9 (number)|9]] iff the sum of its digits is divisible by 9
* sum getæl is dǣlendlic þurh [[10 (number)|10]] iff se endemesta rīm is 0
* sum getæl is dǣlendlic þurh [[11 (number)|11]] iff the alternating sum of its digits is divisible by 11 (e.g. 182919 is divisible by 11 since 1-8+2-9+1-9 = -22 is divisible by 11)





# Ǣlc rīm is tōdǣlendic þurh 1
# A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8.
# Add the digits. The resulting number is divisible by 3 iff the original number was.
# Last 2 digits must be divisible by 4.
# Must end in 0 or 5
# Must be divisible by 2 and 3
# Subtract twice the last digit from the original. The resulting number is divisible by 7 iff the original number was.
# Last 3 digits must be divisible by 8. (If you don't know the divisibility of a 3-digit number by 8, subtract 200 to check to see.)
# Add the digits. The resulting number is divisible by 9 iff the original number was.
# Must end in 0
# Add the digits in odd places and add the digits in even places. The difference between these numbers is divisible by 11 iff the original number was.
# Check divisibility by 3 and 4
# Add 4 times the last digit to the original number. The resulting number is divisible by 13 iff the original number was.
# Check divisibility by 2 and 7
# Check divisibility by 3 and 5
# Last 4 digits must be divisible by 16. (If you don't know the divisibility of a 4-digit number by 16, subtract a multiple of 2000 or 400.)

== Further notions and facts ==

Some elementary rules: 
* If ''a''|''b'' and ''a''|''c'', then ''a''|(''b''+''c''). 
* If ''a''|''b'' and ''b''|''c'', then ''a''|''c''. 
* If ''a''|''b'' and ''b''|''a'', then ''a''=''b'' or ''a''=-''b''.

A positive divisor of ''n'' which is different from ''n'' is called a ''proper divisor'' (or ''aliquot part'') of ''n''. (A number which does not evenly divide ''n', but leaves a remainder, is called an ''aliquant part'' of ''n''.)

An integer ''n'' > 1 whose only proper divisor is 1 is called a [[prime number]]. 

Any positive divisor of ''n'' is a product of [[prime factor|prime divisor]]s of ''n'' raised to some power. This is a consequence of the [[Fundamental theorem of arithmetic]].

If a number equals the sum of its proper divisors, it is said to be a [[perfect number]].  Numbers less than that sum are said to be ''deficient'', while numbers greater than that sum are said to be ''abundant''.

The total number of positive divisors of ''n'' is a [[multiplicative function]] ''d''(''n'') (e.g. ''d''(42) = 8 = 2&times;2&times;2 = ''d''(2)&times;''d''(3)&times;''d''(7)).  The sum of the positive divisors of ''n'' is another multiplicative function &sigma;(''n'') (e.g. &sigma;(42) = 96 = 3&times;4&times;8 = &sigma;(2)&times;&sigma;(3)&times;&sigma;(7)).

The relation | of divisibility turns the set '''N''' of non-negative integers into a [[partially ordered set]], in fact into a [[lattice (order)|complete distributive lattice]]. The largest element of this lattice is 0 and the smallest one is 1. The meet operation ^ is given by the [[greatest common divisor]] and the join operation v by the [[least common multiple]]. This lattice is isomorphic to the dual of the lattice of [[subgroup]]s of the infinite [[cyclic group]] [[integer|'''Z''']].

If an integer ''n'' is written in [[numeral system|base ''b'']], and ''d'' is an integer with ''b'' &equiv; 1 ([[modular arithmetic|mod]] ''d''), then ''n'' is divisible by ''d'' if and only if the sum of its digits is divisible by ''d''. The rules for ''d''=3 and ''d''=9 given above are special cases of this result (''b''=10).

== Generalization ==

One can talk about the concept of divisibility in any [[integral domain]]. Please see that article for the definitions in that setting.

== Divisors in algebraic geometry ==

In [[algebraic geometry]], the word "divisor" is used to mean something rather different. Divisors are a generalization of [[subvariety|subvarieties]] of [[Algebraic variety|algebraic varieties]]; two different generalizations are in common use, [[Cartier divisor]]s and [[Weil divisor]]s.  The concepts agree on nonsingular varieties over algebraically closed fields.  Any Weil divisor is a locally finite [[linear combination]] of irreducible subvarieties of codimension one. To every Cartier divisor '''D''' there is an associated [[line bundle]] denoted by '''[D]''', and the sum of divisors corresponds to [[tensor product]] of line bundles.

== See also==

* [[Table of prime factors]] -- A table of prime factors for 1-1000
* [[Table of divisors]] -- A table of prime and non-prime divisors for 1-1000

== External links ==
* [http://www.farfarfar.com/math/calculators/factoring/ Factoring Calculator] -- Factoring calculator that displays the prime factors and the prime and non-prime divisors of a given number.
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[[Flocc:Rīmcræft]]
An dǣlere is ēac lǣssadǣl of n ȝehāten.

Dǣleras cunnon positife oþþe negatife bēon.  Þā positife dǣleras for 42 sind {1, 2, 3, 6, 7, 14, 21, 42}.

Sum synderlīpe talu: 1 and -1 sind dǣleras for ealle talu, and ȝehƿilċ tæl is dǣlere of 0.  Þā talu þæt 2 dǣlþ ȝehāten efentalu and þā oðrum talum ȝehāten ofertalu.

Flocc:Rīmcræft