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{{Built-in|LCM|∧}} is a [[dyadic]] [[scalar function]] which returns the '''[[wikipedia:Least common multiple|Least Common Multiple]]''' of two integer arguments. It is an extension of [[And]] which maintains the same results on [[Boolean]] arguments and the same [[identity element]] 1, in the same way that [[GCD]] extends [[Or]].
{{Built-in|LCM|∧}} is a [[dyadic]] [[scalar function]] which returns the '''[[wikipedia:Least common multiple|Least Common Multiple]]''' of two integer arguments. It is an extension of [[And]] which maintains the same results on [[Boolean]] arguments and the same [[identity element]] 1, in the same way that [[GCD]] extends [[Or]].


== Extended definition ==
== Examples ==


For positive integer arguments, the least common multiple is the smallest positive number which is divisible by both numbers. If one of the arguments is zero, the LCM function returns zero.
For positive integer arguments, the least common multiple is the smallest positive number which is divisible by both numbers. If one of the arguments is zero, the LCM function returns zero.


<source lang=apl>
<syntaxhighlight lang=apl>
       ∘.∧⍨ 0,⍳10
       ∘.∧⍨ 0,⍳10
0  0  0  0  0  0  0  0  0  0  0
0  0  0  0  0  0  0  0  0  0  0
Line 18: Line 18:
0  9 18  9 36 45 18 63 72  9 90
0  9 18  9 36 45 18 63 72  9 90
0 10 10 30 20 10 30 70 40 90 10
0 10 10 30 20 10 30 70 40 90 10
</source>{{Works in|[[Dyalog APL]]}}
</syntaxhighlight>{{Works in|[[Dyalog APL]]}}


While the mathematical definition of LCM does not cover non-integers, some implementations accept them as arguments. In this case, the return value of <source lang=apl inline>R←X∧Y</source> is chosen so that both <source lang=apl inline>R÷X</source> and <source lang=apl inline>R÷Y</source> are integers (or [[wikipedia:Gaussian integer|Gaussian integers]], when X and/or Y are [[complex]] numbers).
While the mathematical definition of LCM does not cover non-integers, some implementations accept them as arguments. In this case, the return value of <syntaxhighlight lang=apl inline>R←X∧Y</syntaxhighlight> is chosen so that both <syntaxhighlight lang=apl inline>R÷X</syntaxhighlight> and <syntaxhighlight lang=apl inline>R÷Y</syntaxhighlight> are integers (or [[wikipedia:Gaussian integer|Gaussian integers]], when X and/or Y are [[complex]] numbers).


<source lang=apl>
<syntaxhighlight lang=apl>
       0.9∧25÷6
       0.9∧25÷6
112.5
112.5
Line 31: Line 31:
       6J2÷2J2 3J1
       6J2÷2J2 3J1
2J¯1 2
2J¯1 2
</source>{{Works in|[[Dyalog APL]]}}
</syntaxhighlight>{{Works in|[[Dyalog APL]]}}


== Description ==
== Description ==
Line 41: Line 41:
=== Documentation ===
=== Documentation ===


* [http://help.dyalog.com/17.1/#Language/Primitive%20Functions/And%20Lowest%20Common%20Multiple.htm Dyalog]
* [https://help.dyalog.com/17.1/#Language/Primitive%20Functions/And%20Lowest%20Common%20Multiple.htm Dyalog]
* J [https://www.jsoftware.com/help/dictionary/d111.htm Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/stardot#dyadic NuVoc]
* J [https://www.jsoftware.com/help/dictionary/d111.htm Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/stardot#dyadic NuVoc]
{{APL built-ins}}[[Category:Primitive functions]][[Category:Scalar dyadic functions]]
{{APL built-ins}}[[Category:Primitive functions]][[Category:Scalar dyadic functions]]

Latest revision as of 21:28, 10 September 2022

∧

LCM (∧) is a dyadic scalar function which returns the Least Common Multiple of two integer arguments. It is an extension of And which maintains the same results on Boolean arguments and the same identity element 1, in the same way that GCD extends Or.

Examples

For positive integer arguments, the least common multiple is the smallest positive number which is divisible by both numbers. If one of the arguments is zero, the LCM function returns zero. 

      ∘.∧⍨ 0,⍳10
0  0  0  0  0  0  0  0  0  0  0
0  1  2  3  4  5  6  7  8  9 10
0  2  2  6  4 10  6 14  8 18 10
0  3  6  3 12 15  6 21 24  9 30
0  4  4 12  4 20 12 28  8 36 20
0  5 10 15 20  5 30 35 40 45 10
0  6  6  6 12 30  6 42 24 18 30
0  7 14 21 28 35 42  7 56 63 70
0  8  8 24  8 40 24 56  8 72 40
0  9 18  9 36 45 18 63 72  9 90
0 10 10 30 20 10 30 70 40 90 10
Works in: Dyalog APL

While the mathematical definition of LCM does not cover non-integers, some implementations accept them as arguments. In this case, the return value of R←X∧Y is chosen so that both R÷X and R÷Y are integers (or Gaussian integers, when X and/or Y are complex numbers). 

      0.9∧25÷6
112.5
      112.5÷0.9(25÷6)
125 27
      2J2∧3J1
6J2
      6J2÷2J2 3J1
2J¯1 2
Works in: Dyalog APL

Description

The LCM of two numbers is their product divided by the GCD.

External links

Documentation

APL built-ins [edit]
Primitives (Timeline) Functions
Scalar
Monadic Conjugate ∙ Negate ∙ Signum ∙ Reciprocal ∙ Magnitude ∙ Exponential ∙ Natural Logarithm ∙ Floor ∙ Ceiling ∙ Factorial ∙ Not ∙ Pi Times ∙ Roll ∙ Type ∙ Imaginary ∙ Square Root ∙ Round
Dyadic Add ∙ Subtract ∙ Times ∙ Divide ∙ Residue ∙ Power ∙ Logarithm ∙ Minimum ∙ Maximum ∙ Binomial ∙ Comparison functions ∙ Boolean functions (And, Or, Nand, Nor) ∙ GCD ∙ LCM ∙ Circular ∙ Complex ∙ Root
Non-Scalar
Structural Shape ∙ Reshape ∙ Tally ∙ Depth ∙ Ravel ∙ Enlist ∙ Table ∙ Catenate ∙ Reverse ∙ Rotate ∙ Transpose ∙ Raze ∙ Mix ∙ Split ∙ Enclose ∙ Nest ∙ Cut (K) ∙ Pair ∙ Link ∙ Partitioned Enclose ∙ Partition
Selection First ∙ Pick ∙ Take ∙ Drop ∙ Unique ∙ Identity ∙ Stop ∙ Select ∙ Replicate ∙ Expand ∙ Set functions (Intersection ∙ Union ∙ Without) ∙ Bracket indexing ∙ Index ∙ Cartesian Product ∙ Sort
Selector Index generator ∙ Grade ∙ Index Of ∙ Interval Index ∙ Indices ∙ Deal ∙ Prefix and suffix vectors
Computational Match ∙ Not Match ∙ Membership ∙ Find ∙ Nub Sieve ∙ Encode ∙ Decode ∙ Matrix Inverse ∙ Matrix Divide ∙ Format ∙ Execute ∙ Materialise ∙ Range
Operators Monadic Each ∙ Commute ∙ Constant ∙ Replicate ∙ Expand ∙ Reduce ∙ Windowed Reduce ∙ Scan ∙ Outer Product ∙ Key ∙ I-Beam ∙ Spawn ∙ Function axis ∙ Identity (Null, Ident)
Dyadic Bind ∙ Compositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner Product ∙ Determinant ∙ Power ∙ At ∙ Under ∙ Rank ∙ Depth ∙ Variant ∙ Stencil ∙ Cut ∙ Direct definition (operator) ∙ Identity (Lev, Dex)
Quad names Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector
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