LCM: Difference between revisions

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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.

Revision as of 02:44, 2 June 2020

∧

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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