Smallest multiple

Smallest multiple is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Task

Task description is taken from Project Euler
(https://projecteuler.net/problem=5)
2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.
What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?

Related

• Least common multiple

11l

F f(n)
   V ans = BigInt(1)
   L(i) 1..n
      ans *= BigInt(i) I/ gcd(BigInt(i), BigInt(ans))
   R ans

L(n) [10, 20, 200, 2000]
   print(n‘: ’f(n))

Output:

10: 2520
20: 232792560
200: 337293588832926264639465766794841407432394382785157234228847021917234018060677390066992000
2000: 151117794877444315307536308337572822173736308853579339903227904473000476322347234655122160866668946941993951014270933512030194957221371956828843521568082173786251242333157830450435623211664308500316844478617809101158220672108895053508829266120497031742749376045929890296052805527212315382805219353316270742572401962035464878235703759464796806075131056520079836955770415021318508272982103736658633390411347759000563271226062182345964184167346918225243856348794013355418404695826256911622054015423611375261945905974225257659010379414787547681984112941581325198396634685659217861208771400322507388161967513719166366839894214040787733471287845629833993885413462225294548785581641804620417256563685280586511301918399010451347815776570842790738545306707750937624267501103840324470083425714138183905657667736579430274197734179172691637931540695631396056193786415805463680000

ALGOL 68

Translation of: Wren

Works with: ALGOL 68G version Any - tested with release 2.8.3.win32

Uses Algol 68G's LONG LONG INT which has specifiable precision.

Library: ALGOL 68-primes

BEGIN # find the smallest number that is divisible by each of the numbers 1..n #
      # translation of the Wren sample #
    PR precision 1000 PR # set the precision of LONG LONG INT #
    PR read "primes.incl.a68" PR
    # returns the lowest common multiple of the numbers 1 : n #
    PROC lcm = ( INT n )LONG LONG INT:
         BEGIN
            # sieve the primes to n #
            []BOOL prime = PRIMESIEVE n;
            LONG LONG INT result := 1;
            FOR p TO UPB prime DO
                IF prime[ p ] THEN          
                    LONG LONG INT f := p;           # f will be set to the #
                    WHILE f * p <= n DO f *:= p OD; # highest multiple of p <= n #
                    result *:= f
                FI
            OD;
            result
         END # lcm # ;
    # returns a string representation of n with commas #
    PROC commatise = ( LONG LONG INT n )STRING:
         BEGIN
            STRING result      := "";
            STRING unformatted  = whole( n, 0 );
            INT    ch count    := 0;
            FOR c FROM UPB unformatted BY -1 TO LWB unformatted DO
                IF   ch count <= 2 THEN ch count +:= 1
                ELSE                    ch count  := 1; "," +=: result
                FI;
                unformatted[ c ] +=: result
            OD;
            result
         END; # commatise #
    print( ( "The LCMs of the numbers 1 to N inclusive is:", newline ) );
    []INT tests = ( 10, 20, 200, 2000 );
    FOR i FROM LWB tests TO UPB tests DO
        print( ( whole( tests[ i ], -5 ), ": ", commatise( lcm( tests[ i ] ) ), newline ) )
    OD
END

Output:

   10: 2,520
   20: 232,792,560
  200: 337,293,588,832,926,264,639,465,766,794,841,407,432,394,382,785,157,234,228,847,021,917,234,018,060,677,390,066,992,000
 2000: 151,117,794,877,444,315,307,536,308,337,572,822,173,736,308,853,579,339,903,227,904,473,000,476,322,347,234,655,122,160,866,668,946,941,993,951,014,270,933,512,030,194,957,221,371,956,828,843,521,568,082,173,786,251,242,333,157,830,450,435,623,211,664,308,500,316,844,478,617,809,101,158,220,672,108,895,053,508,829,266,120,497,031,742,749,376,045,929,890,296,052,805,527,212,315,382,805,219,353,316,270,742,572,401,962,035,464,878,235,703,759,464,796,806,075,131,056,520,079,836,955,770,415,021,318,508,272,982,103,736,658,633,390,411,347,759,000,563,271,226,062,182,345,964,184,167,346,918,225,243,856,348,794,013,355,418,404,695,826,256,911,622,054,015,423,611,375,261,945,905,974,225,257,659,010,379,414,787,547,681,984,112,941,581,325,198,396,634,685,659,217,861,208,771,400,322,507,388,161,967,513,719,166,366,839,894,214,040,787,733,471,287,845,629,833,993,885,413,462,225,294,548,785,581,641,804,620,417,256,563,685,280,586,511,301,918,399,010,451,347,815,776,570,842,790,738,545,306,707,750,937,624,267,501,103,840,324,470,083,425,714,138,183,905,657,667,736,579,430,274,197,734,179,172,691,637,931,540,695,631,396,056,193,786,415,805,463,680,000

Arturo

print first select.first range.step:20 20 ∞ 'x ->
    every? 11..19 'z -> zero? x % z

Output:

232792560

Asymptote

int temp = 2*3*5*7*11*13*17*19;
int smalmul = temp;
int lim = 1;
while (lim <= 20) {
	lim = lim + 1;
	while (smalmul % lim != 0) {
      lim = 1;
      smalmul = smalmul + temp;
    }
}
write(smalmul);

AutoHotkey

primes := 1
loop 20
    if prime_numbers(A_Index).Count() = 1
        primes *= A_Index

loop
{
    Result := A_Index*primes
    loop 20
        if Mod(Result, A_Index)
            continue, 2
    break
}
MsgBox % Result
return

prime_numbers(n) { ; http://www.rosettacode.org/wiki/Prime_decomposition#Optimized_Version
    if (n <= 3)
        return [n]
    ans := [], done := false
    while !done 
    {
        if !Mod(n,2){
            ans.push(2), n /= 2
            continue
        }
        if !Mod(n,3) {
            ans.push(3), n /= 3
            continue
        }
        if (n = 1)
            return ans
        sr := sqrt(n), done := true
        ; try to divide the checked number by all numbers till its square root.
        i := 6
        while (i <= sr+6){
            if !Mod(n, i-1) { ; is n divisible by i-1?
                ans.push(i-1), n /= i-1, done := false
                break
            }
            if !Mod(n, i+1) { ; is n divisible by i+1?
                ans.push(i+1), n /= i+1, done := false
                break
            }
            i += 6
        }
    }
    ans.push(n)
    return ans
}

Output:

232792560

BASIC

BASIC256

temp = 2*3*5*7*11*13*17*19
smalmul = temp
lim = 1
do
    lim += 1
    if (smalmul mod lim) then lim = 1 : smalmul += temp
until lim = 20
print smalmul

Output:

232792560

PureBasic

OpenConsole()
temp.i = 2*3*5*7*11*13*17*19
smalmul.i = temp
lim.i = 1
Repeat
  lim + 1
  If (smalmul % lim)
    lim = 1 
    smalmul = smalmul + temp
  EndIf
Until lim = 20
PrintN(Str(smalmul))
Input()
CloseConsole()

Output:

232792560

True BASIC

LET temp = 2*3*5*7*11*13*17*19
LET smalmul = temp
LET lim = 1
DO
   LET lim = lim+1
   IF (REMAINDER(ROUND(smalmul),ROUND(lim)) <> 0) THEN
      LET lim = 1
      LET smalmul = smalmul+temp
   END IF
LOOP UNTIL lim = 20
PRINT smalmul
END

Output:

232792560

Delphi

Works with: Delphi version 6.0

Library: SysUtils,StdCtrls

function IsDivisible120(N: integer): boolean;
{Is N evenly divisible by numbers 1..20}
var I: integer;
begin
Result:=False;
{For speed - larger numbers less likely divisor}
for I:=20 downto 2 do
 if (N mod I)<>0 then exit;
Result:=True;
end;


procedure SmallestDivide120(Memo: TMemo);
var I: integer;
begin
{Only look at even numbers for speed}
for I:=1 to High(Integer) do
 if IsDivisible120(I*2) then
	begin
	Memo.Lines.Add(FloatToStrF(I*2,ffNumber,18,0));
	break;
	end;
end;

Output:

232,792,560
Elapsed Time: 920.406 ms.

F#

This task uses Extensible Prime Generator (F#)

// Least Multiple. Nigel Galloway: October 22nd., 2021
let fG n g=let rec fN i=match i*g with g when n>g->fN g |_->i in fN g
let leastMult n=let fG=fG n in primes32()|>Seq.takeWhile((>=)n)|>Seq.map fG|>Seq.reduce((*))
printfn $"%d{leastMult 20}"

Output:

232792560

Factor

Works with: Factor version 0.98

USING: math.functions math.ranges prettyprint sequences ;

20 [1,b] 1 [ lcm ] reduce .

Output:

232792560

Fermat

Func Ilog( n, b ) =
    i:=0;                  {integer logarithm of n to base b, positive only}
    while b^i<=n do
        i:+;
    od;
    i-1.;
    
Func Smalmul( n ) =
    s:=1;
    for a = 1 to n do
        if Isprime(a) then s:=s*a^Ilog(n, a) fi;
    od;
    s.;

!Smalmul(20);

Output:

232792560

FreeBASIC

Use the code from the Least common multiple example as an include.

#include"lcm.bas"

redim shared as ulongint smalls(0 to 1)  'calculate and store as we go
smalls(0) = 0: smalls(1) = 1

function smalmul(n as longint) as ulongint
    if n<0 then return smalmul(-n)     'deal with negative input
    dim as uinteger m = ubound(smalls)
    if n<=m then return smalls(n)  'have we calculated this already
    'if not, make room for the next bunch of terms
    redim preserve as ulongint smalls(0 to n)
    for i as uinteger = m+1 to n
        smalls(i) = lcm(smalls(i-1), i)
    next i
    return smalls(n)
end function

for i as uinteger = 0 to 20
    print i, smalmul(i)
next i

Go

Translation of: Wren

Library: Go-rcu

package main

import (
    "fmt"
    "math/big"
    "rcu"
)

func lcm(n int) *big.Int {
    lcm := big.NewInt(1)
    t := new(big.Int)
    for _, p := range rcu.Primes(n) {
        f := p
        for f*p <= n {
            f *= p
        }
        lcm.Mul(lcm, t.SetUint64(uint64(f)))
    }
    return lcm
}

func main() {
    fmt.Println("The LCMs of the numbers 1 to N inclusive is:")
    for _, i := range []int{10, 20, 200, 2000} {
        fmt.Printf("%4d: %s\n", i, lcm(i))
    }
}

Output:

The LCMs of the numbers 1 to N inclusive is:
  10: 2520
  20: 232792560
 200: 337293588832926264639465766794841407432394382785157234228847021917234018060677390066992000
2000: 151117794877444315307536308337572822173736308853579339903227904473000476322347234655122160866668946941993951014270933512030194957221371956828843521568082173786251242333157830450435623211664308500316844478617809101158220672108895053508829266120497031742749376045929890296052805527212315382805219353316270742572401962035464878235703759464796806075131056520079836955770415021318508272982103736658633390411347759000563271226062182345964184167346918225243856348794013355418404695826256911622054015423611375261945905974225257659010379414787547681984112941581325198396634685659217861208771400322507388161967513719166366839894214040787733471287845629833993885413462225294548785581641804620417256563685280586511301918399010451347815776570842790738545306707750937624267501103840324470083425714138183905657667736579430274197734179172691637931540695631396056193786415805463680000

Haskell

import Text.Printf (printf)

--- SMALLEST INTEGER EVENLY DIVISIBLE BY EACH OF [1..N] --

smallest :: Integer -> Integer
smallest =
  foldr lcm 1 . enumFromTo 1


--------------------------- TEST -------------------------
main :: IO ()
main =
  (putStrLn . unlines) $
    showSmallest <$> [10, 20, 200, 2000]

------------------------- DISPLAY ------------------------
showSmallest :: Integer -> String
showSmallest =
  ((<>) . (<> " -> ") . printf "%4d")
    <*> (printf "%d" . smallest)

Output:

  10 -> 2520
  20 -> 232792560
 200 -> 337293588832926264639465766794841407432394382785157234228847021917234018060677390066992000
2000 -> 151117794877444315307536308337572822173736308853579339903227904473000476322347234655122160866668946941993951014270933512030194957221371956828843521568082173786251242333157830450435623211664308500316844478617809101158220672108895053508829266120497031742749376045929890296052805527212315382805219353316270742572401962035464878235703759464796806075131056520079836955770415021318508272982103736658633390411347759000563271226062182345964184167346918225243856348794013355418404695826256911622054015423611375261945905974225257659010379414787547681984112941581325198396634685659217861208771400322507388161967513719166366839894214040787733471287845629833993885413462225294548785581641804620417256563685280586511301918399010451347815776570842790738545306707750937624267501103840324470083425714138183905657667736579430274197734179172691637931540695631396056193786415805463680000

J

   *./ >: i. 20
232792560

jq

Works with jq (*)
Works with gojq, the Go implementation of jq

The following uses `is_prime` as defined at Erdős-primes#jq.

(*) The C implementation of jq has sufficient accuracy for N == 20 but not N == 200, so the output shown below is based on a run of gojq.

# Output: a stream of primes less than $n in increasing order
def primes($n):
  2, (range(3; $n; 2) | select(is_prime));

# lcm of 1 to $n inclusive
def lcm:
  . as $n
  | reduce primes($n) as $p (1;
      . * ($p | until(. * $p > $n; . * $p)) ) ;

"N: LCM of the numbers 1 to N inclusive",
 ( 10, 20, 200, 2000
   | "\(.): \(smallest_multiple)" )

Output:

N: LCM of the numbers 1 to N inclusive
10: 2520
20: 232792560
200: 337293588832926264639465766794841407432394382785157234228847021917234018060677390066992000
2000: 151117794877444315307536308337572822173736308853579339903227904473000476322347234655122160866668946941993951014270933512030194957221371956828843521568082173786251242333157830450435623211664308500316844478617809101158220672108895053508829266120497031742749376045929890296052805527212315382805219353316270742572401962035464878235703759464796806075131056520079836955770415021318508272982103736658633390411347759000563271226062182345964184167346918225243856348794013355418404695826256911622054015423611375261945905974225257659010379414787547681984112941581325198396634685659217861208771400322507388161967513719166366839894214040787733471287845629833993885413462225294548785581641804620417256563685280586511301918399010451347815776570842790738545306707750937624267501103840324470083425714138183905657667736579430274197734179172691637931540695631396056193786415805463680000

Julia

julia> foreach(x -> @show(lcm(x)), [1:10, 1:20, big"1":200, big"1":2000])
lcm(x) = 2520
lcm(x) = 232792560
lcm(x) = 337293588832926264639465766794841407432394382785157234228847021917234018060677390066992000
lcm(x) = 151117794877444315307536308337572822173736308853579339903227904473000476322347234655122160866668946941993951014270933512030194957221371956828843521568082173786251242333157830450435623211664308500316844478617809101158220672108895053508829266120497031742749376045929890296052805527212315382805219353316270742572401962035464878235703759464796806075131056520079836955770415021318508272982103736658633390411347759000563271226062182345964184167346918225243856348794013355418404695826256911622054015423611375261945905974225257659010379414787547681984112941581325198396634685659217861208771400322507388161967513719166366839894214040787733471287845629833993885413462225294548785581641804620417256563685280586511301918399010451347815776570842790738545306707750937624267501103840324470083425714138183905657667736579430274197734179172691637931540695631396056193786415805463680000

Mathematica / Wolfram Language

LCM @@ Range[20]

Output:

232792560	

OCaml

let rec gcd a = function
  | 0 -> a
  | b -> gcd b (a mod b)

let lcm a b =
  a * b / gcd a b

let smallest_multiple n =
  Seq.(ints 1 |> take n |> fold_left lcm 1)

let () =
  Printf.printf "%u\n" (smallest_multiple 20)

Output:

232792560

Pascal

Here the simplest way, like Raku, check the highest exponent of every prime in range
Using harded coded primes.

{$IFDEF FPC}
  {$MODE DELPHI}
{$ELSE}
  {$APPTAYPE CONSOLE}
{$ENDIF}
const
 smallprimes : array[0..10] of Uint32 = (2,3,5,7,11,13,17,19,23,29,31);
 MAX = 20;

function getmaxfac(pr: Uint32): Uint32;
//get the pr^highest exponent of prime used in 2 .. MAX
var
  i,fac : integer;
Begin
  result := pr;
  while pr*result <= MAX do
    result *= pr;
end;

var
  n,pr,prIdx : Uint32;
BEGIN
  n := 1;
  prIdx := 0;
  pr := smallprimes[prIdx];
  repeat
    pr := smallprimes[prIdx];
    n *= getmaxfac(pr);
    inc(prIdx);
    pr := smallprimes[prIdx];
  until pr>MAX;
  writeln(n);
{$IFDEF WINDOWS}
  READLN;
{$ENDIF}
END.

Output:

  232792560

extended

fascinating find, that the count of digits is nearly a constant x upper rangelimit.
The number of factors is the count of primes til limit.See GetFactorList.
No need for calculating lcm(lcm(lcm(1,2),3),4..) or prime decomposition
Using prime sieve.

{$IFDEF FPC}
  {$MODE DELPHI} {$Optimization On}
{$ELSE}
  {$APPTAYPE CONSOLE}
{$ENDIF}
{$DEFINE USE_GMP}
uses
  {$IFDEF USE_GMP}
  gmp,
  {$ENDIF}
  sysutils; //format
const
  MAX_LIMIT = 2*1000*1000;
  UpperLimit = MAX_LIMIT+1000;// so to find a prime beyond MAX_LIMIT
  MAX_UINT64 = 46;// unused.Limit to get an Uint64 output
type
  tFactors = array of Uint32;
  tprimelist = array of byte;
var
  primeDeltalist : tPrimelist;
  factors,
  saveFactors:tFactors;
  saveFactorsIdx,
  maxFactorsIdx : Uint32;
procedure Init_Primes;
var
  pPrime : pByte;
  p,i,delta,cnt: NativeUInt;
begin
  setlength(primeDeltalist,UpperLimit+3*8+1);
  pPrime := @primeDeltalist[0];
  //delete multiples of 2,3
  i := 0;
  repeat
    //take care of endianess //0706050403020100
    pUint64(@pPrime[i+0])^ := $0100010000000100;
    pUint64(@pPrime[i+8])^ := $0000010001000000;
    pUint64(@pPrime[i+16])^:= $0100000001000100;
    inc(i,24);
  until i>UpperLimit;
  cnt := 2;// 2,3
  p := 5;
  delta := 1;//5-3
  repeat
    if pPrime[p] <> 0 then
    begin
      i := p*p;
      if i > UpperLimit then
        break;
      inc(cnt);
      pPrime[p-2*delta] := delta;
      delta := 0;
      repeat
        pPrime[i] := 0;
        inc(i,2*p);
      until i>UpperLimit;
    end;
    inc(p,2);
    inc(delta);
  until p*p>UpperLimit;
  setlength(saveFactors,cnt);
  //convert to delta
  repeat
    if pPrime[p]<> 0 then
    begin
      pPrime[p-2*delta] := delta;
      inc(cnt);
      delta := 0;
    end;
    inc(p,2);
    inc(delta);
  until p > UpperLimit;
  setlength(factors,cnt);
  factors[0] := 2;
  factors[1] := 3;
  i := 2;
  p := 5;
  repeat
    factors[i] := p;
    p += 2*pPrime[p];
    i += 1;
  until i >= cnt;
  setlength(primeDeltalist,0);
//  writeln(length(savefactors)); writeln(length(factors));
end;

{$IFDEF USE_GMP}
procedure ConvertToMPZ(const factors:tFactors;dgtCnt:UInt32);
const
  c19Digits = QWord(10*1000000)*1000000*1000000;
var
  mp,mpdiv : mpz_t;
  s : AnsiString;
  rest,last : Uint64;
  f : Uint32;
  i :int32;
begin
  //Init and allocate space
  mpz_init_set_ui(mp,0);
  mpz_init(mpdiv);
  mpz_ui_pow_ui(mpdiv,10,dgtCnt);
  mpz_add(mp,mp,mpdiv);
  mpz_add_ui(mp,mp,1);
  mpz_set_ui(mp,1);

  i := maxFactorsIdx;
  rest := 1;
  repeat
    last := rest;
    f := factors[i];
    rest *= f;
    if rest div f <> last then
    begin
      mpz_mul_ui(mp,mp,last);
      rest := f;
    end;
    dec(i);
  until i < 0;
  mpz_mul_ui(mp,mp,rest);

  If dgtcnt>40 then
  begin
    rest := mpz_fdiv_ui(mp,c19Digits);
    s := '..'+Format('%.19u',[rest]);
    mpz_fdiv_q_ui (mpdiv,mpdiv,c19Digits);
    mpz_fdiv_q(mp,mp,mpdiv);
    rest := mpz_get_ui(mp);
    writeln(rest:19,s);
    mpz_clear(mpdiv);
  end
  else
  Begin
    setlength(s,dgtCnt+1000);
    mpz_get_str(@s[1],10,mp);
    writeln(s);
    i := length(s);
    while not(s[i] in['0'..'9']) do
      dec(i);
    setlength(s,i+1);
    writeln(s);
  end;
  mpz_clear(mp);
end;
{$ENDIF}

procedure CheckDigits(const factors:tFactors);
var
  dgtcnt : extended;
  i : integer;
begin
  dgtcnt := 0;
  i := 0;
  repeat
    dgtcnt += ln(factors[i]);
    inc(i);
  until i > maxFactorsIdx;
  dgtcnt := trunc(dgtcnt/ln(10))+1;
  writeln(' has ',maxFactorsIdx+1:10,' factors and ',dgtcnt:10:0,' digits');
  {$IFDEF USE_GMP}
    i := trunc(dgtcnt);
    if i < 1000*1000 then
      ConvertToMPZ(factors,i);
  {$ENDIF}
end;

function ConvertToUint64(const factors:tFactors):Uint64;
var
  i : integer;
begin
  if maxFactorsIdx >15 then
    Exit(0);
  result := 1;
  for i := 0 to maxFactorsIdx do
    result *= factors[i];
end;

function ConvertToStr(const factors:tFactors):Ansistring;
var
  s : Ansistring;
  i : integer;
begin
  result := '';
  for i := 0 to maxFactorsIdx-1 do
  begin
    str(factors[i],s);
    result += s+'*';
  end;
  str(factors[maxFactorsIdx],s);
  result += s;
end;

procedure GetFactorList(var factors:tFactors;max:Uint32);
var
  p,f,lf : Uint32;
BEGIN
  p := 2;
  lf := 0;
  saveFactors[lf] := p;
  while p*p <= max do
  Begin
    saveFactors[lf] := p;
    f := p*p;
    while f*p <= max do
      f*= p;
    factors[lf] := f;
    inc(lf);
    p := factors[lf];
    if p= 0 then HALT;
  end;
  if lf>0 then
    saveFactorsIdx := lf-1;
  repeat
    inc(lf)
  until factors[lf]>Max;
  maxFactorsIdx := lf-1;
end;

procedure Check(var factors:tFactors;max:Uint32);
var
  i: Uint32;
begin
  GetFactorList(factors,max);
  write(max:10,': ');
  if maxFactorsIdx>15 then
    CheckDigits(factors)
  else
    writeln(ConvertToUint64(factors):21,' = ',ConvertToStr(factors));
  for i := 0 to saveFactorsIdx do
    factors[i] := savefactors[i];
end;

var
  max: Uint32;
BEGIN
  Init_Primes;

  max := 2;
  repeat
    check(factors,max);
    max *=10;
  until max > MAX_LIMIT;

  writeln;
  For max := 10 to 20 do // < MAX_UINT64
    check(factors,max);
{$IFDEF WINDOWS}
  READLN;
{$ENDIF}
END.

Output:

TIO.RUN Real time: 1.161 s User time: 1.106 s Sys. time: 0.049 s CPU share: 99.49 %
         2:                     2 = 2
        20:             232792560 = 16*9*5*7*11*13*17*19
       200:  has         46 factors and         90 digits
3372935888329262646..8060677390066992000
      2000:  has        303 factors and        867 digits
1511177948774443153..3786415805463680000
     20000:  has       2262 factors and       8676 digits
4879325627288270518..7411295098112000000
    200000:  has      17984 factors and      86871 digits
3942319728529926377..9513860925440000000
   2000000:  has     148933 factors and     868639 digits
8467191629995920178..6480233472000000000
{ at home
  20000000:  has    1270607 factors and    8686151 digits
1681437413936981958..6706037760000000000
 200000000:  has   11078937 factors and   86857606 digits
2000000000:  has   98222287 factors and  868583388 digits
}
        10:                  2520 = 8*9*5*7
        11:                 27720 = 8*9*5*7*11
        12:                 27720 = 8*9*5*7*11
        13:                360360 = 8*9*5*7*11*13
        14:                360360 = 8*9*5*7*11*13
        15:                360360 = 8*9*5*7*11*13
        16:                720720 = 16*9*5*7*11*13
        17:              12252240 = 16*9*5*7*11*13*17
        18:              12252240 = 16*9*5*7*11*13*17
        19:             232792560 = 16*9*5*7*11*13*17*19
        20:             232792560 = 16*9*5*7*11*13*17*19

Perl

#!/usr/bin/perl

use strict; # https://rosettacode.org/wiki/Smallest_multiple#Raku
use warnings;
use ntheory qw( lcm );

print "for $_, it's @{[ lcm(1 .. $_) ]}\n" for 10, 20;

Output:

for 10, it's 2520
for 20, it's 232792560

Phix

Using the builtin, limited to 2⁵³ aka N=36 on 32-bit, 2⁶⁴ aka N=46 on 64-bit.

with javascript_semantics
?lcm(tagset(20))

Output:

232792560

Using gmp

Translation of: Wren

with javascript_semantics
include mpfr.e
procedure plcmz(integer n)
    sequence primes = get_primes_le(n)
    mpz res = mpz_init(1)
    for i=1 to length(primes) do
        integer p = primes[i], f = p
        while f*p <= n do f *= p end while
        mpz_mul_si(res,res,f)
    end for
    printf(1,"%,5d: %s\n", {n, shorten(mpz_get_str(res,10,true))})
end procedure
 
printf(1,"The LCMs of the numbers 1 to N inclusive is:\n")
papply({10,20,200,2000},plcmz)

Output:

The LCMs of the numbers 1 to N inclusive is:
   10: 2,520
   20: 232,792,560
  200: 337,293,588,832,926,...,677,390,066,992,000 (90 digits)
2,000: 151,117,794,877,444,...,415,805,463,680,000 (867 digits)

Picat

lcm/2

lcm/2 is defined as:

lcm(X,Y) = X*Y//gcd(X,Y).

Iteration

smallest_multiple_range1(N) = A =>
  A = 1,
  foreach(E in 2..N) 
    A := lcm(A,E)
  end.

fold/3

smallest_multiple_range2(N) = fold(lcm, 1, 2..N).

reduce/2

smallest_multiple_range3(N) = reduce(lcm, 2..N).

Testing

Of the three implementations the fold/3 approach is slightly faster than the other two.

main =>
   foreach(N in [10,20,200,2000])
     println(N=smallest_multiple_range2(N))
   end.

Output:

10 = 2520
20 = 232792560
200 = 337293588832926264639465766794841407432394382785157234228847021917234018060677390066992000
2000 = 151117794877444315307536308337572822173736308853579339903227904473000476322347234655122160866668946941993951014270933512030194957221371956828843521568082173786251242333157830450435623211664308500316844478617809101158220672108895053508829266120497031742749376045929890296052805527212315382805219353316270742572401962035464878235703759464796806075131056520079836955770415021318508272982103736658633390411347759000563271226062182345964184167346918225243856348794013355418404695826256911622054015423611375261945905974225257659010379414787547681984112941581325198396634685659217861208771400322507388161967513719166366839894214040787733471287845629833993885413462225294548785581641804620417256563685280586511301918399010451347815776570842790738545306707750937624267501103840324470083425714138183905657667736579430274197734179172691637931540695631396056193786415805463680000

Python

""" Rosetta code task: Smallest_multiple """

from math import gcd
from functools import reduce


def lcm(a, b):
    """ least common multiple """
    return 0 if 0 == a or 0 == b else (
        abs(a * b) // gcd(a, b)
    )


for i in [10, 20, 200, 2000]:
    print(str(i) + ':', reduce(lcm, range(1, i + 1)))

Output:

10: 2520
20: 232792560
200: 337293588832926264639465766794841407432394382785157234228847021917234018060677390066992000
2000: 151117794877444315307536308337572822173736308853579339903227904473000476322347234655122160866668946941993951014270933512030194957221371956828843521568082173786251242333157830450435623211664308500316844478617809101158220672108895053508829266120497031742749376045929890296052805527212315382805219353316270742572401962035464878235703759464796806075131056520079836955770415021318508272982103736658633390411347759000563271226062182345964184167346918225243856348794013355418404695826256911622054015423611375261945905974225257659010379414787547681984112941581325198396634685659217861208771400322507388161967513719166366839894214040787733471287845629833993885413462225294548785581641804620417256563685280586511301918399010451347815776570842790738545306707750937624267501103840324470083425714138183905657667736579430274197734179172691637931540695631396056193786415805463680000

Quackery

lcm is defined at Least common multiple#Quackery.

  [ 1 swap times [ i 1+ lcm ] ] is smalmul ( n --> n )

  ' [ 10 20 200 2000 ] witheach [ dup echo say ": " smalmul echo cr ]

Output:

10: 2520
20: 232792560
200: 337293588832926264639465766794841407432394382785157234228847021917234018060677390066992000
2000: 151117794877444315307536308337572822173736308853579339903227904473000476322347234655122160866668946941993951014270933512030194957221371956828843521568082173786251242333157830450435623211664308500316844478617809101158220672108895053508829266120497031742749376045929890296052805527212315382805219353316270742572401962035464878235703759464796806075131056520079836955770415021318508272982103736658633390411347759000563271226062182345964184167346918225243856348794013355418404695826256911622054015423611375261945905974225257659010379414787547681984112941581325198396634685659217861208771400322507388161967513719166366839894214040787733471287845629833993885413462225294548785581641804620417256563685280586511301918399010451347815776570842790738545306707750937624267501103840324470083425714138183905657667736579430274197734179172691637931540695631396056193786415805463680000

Raku

Exercise with some larger values as well.

say "$_: ", [lcm] 2..$_ for <10 20 200 2000>

Output:

10: 2520
20: 232792560
200: 337293588832926264639465766794841407432394382785157234228847021917234018060677390066992000
2000: 151117794877444315307536308337572822173736308853579339903227904473000476322347234655122160866668946941993951014270933512030194957221371956828843521568082173786251242333157830450435623211664308500316844478617809101158220672108895053508829266120497031742749376045929890296052805527212315382805219353316270742572401962035464878235703759464796806075131056520079836955770415021318508272982103736658633390411347759000563271226062182345964184167346918225243856348794013355418404695826256911622054015423611375261945905974225257659010379414787547681984112941581325198396634685659217861208771400322507388161967513719166366839894214040787733471287845629833993885413462225294548785581641804620417256563685280586511301918399010451347815776570842790738545306707750937624267501103840324470083425714138183905657667736579430274197734179172691637931540695631396056193786415805463680000

Ring

see "working..." + nl
see "Smallest multiple is:" + nl
n = 0

while true
      n++
      flag = 0
      for m = 1 to 20
          if n % m = 0
             flag += 1
          ok
      next
      if flag = 20
         see "" + n + nl
         exit
      ok
end

see "done..." + nl

Output:

working...
Smallest multiple is:
232792560
done...

RPL

Translation of: BASIC

≪ 2 3 * 5 * 7 * 9 * 11 * 13 * 17 * 19 * → t
  ≪ t 2 20 FOR lim 
        IF DUP lim MOD THEN 1 'lim' STO t + END NEXT
≫ ≫ 'TASK' STO

With LCM defined at Least common multiple:

≪ 1 2 20 FOR n n LCM NEXT ≫ 'TASK' STO

Output:

1: 232792560

Ruby

[10, 20, 200, 2000].each {|n| puts "#{n}: #{(1..n).inject(&:lcm)}" }

Output:

10: 2520
20: 232792560
200: 337293588832926264639465766794841407432394382785157234228847021917234018060677390066992000
2000: 151117794877444315307536308337572822173736308853579339903227904473000476322347234655122160866668946941993951014270933512030194957221371956828843521568082173786251242333157830450435623211664308500316844478617809101158220672108895053508829266120497031742749376045929890296052805527212315382805219353316270742572401962035464878235703759464796806075131056520079836955770415021318508272982103736658633390411347759000563271226062182345964184167346918225243856348794013355418404695826256911622054015423611375261945905974225257659010379414787547681984112941581325198396634685659217861208771400322507388161967513719166366839894214040787733471287845629833993885413462225294548785581641804620417256563685280586511301918399010451347815776570842790738545306707750937624267501103840324470083425714138183905657667736579430274197734179172691637931540695631396056193786415805463680000

Verilog

Translation of: Yabasic

module main;
    integer temp, smalmul, lim;
  
  initial begin
    temp = 2*3*5*7*11*13*17*19;
    smalmul = temp;
    lim = 1;
    
    while (lim <= 20) begin
	lim = lim + 1;
	while (smalmul % lim != 0) begin
          lim = 1;
          smalmul = smalmul + temp;
        end
    end

      $display(smalmul);
      $finish ;
    end
endmodule

Output:

232792560

Wren

Library: Wren-math

Library: Wren-big

Library: Wren-fmt

We don't really need a computer for the task as set because it's just the product of the maximum prime powers <= 20 which is : 16 x 9 x 5 x 7 x 11 x 13 x 17 x 19 = 232,792,560.

More formally and quite quick by Wren standards at 0.017 seconds:

import "./math" for Int
import "./big" for BigInt
import "./fmt" for Fmt

var lcm = Fn.new { |n|
    var primes = Int.primeSieve(n)
    var lcm = BigInt.one
    for (p in primes) {
        var f = p
        while (f * p <= n) f = f * p
        lcm = lcm * f
    }
    return lcm
}

System.print("The LCMs of the numbers 1 to N inclusive is:")
for (i in [10, 20, 200, 2000]) Fmt.print("$,5d: $,i", i, lcm.call(i))

Output:

The LCMs of the numbers 1 to N inclusive is:
   10: 2,520
   20: 232,792,560
  200: 337,293,588,832,926,264,639,465,766,794,841,407,432,394,382,785,157,234,228,847,021,917,234,018,060,677,390,066,992,000
2,000: 151,117,794,877,444,315,307,536,308,337,572,822,173,736,308,853,579,339,903,227,904,473,000,476,322,347,234,655,122,160,866,668,946,941,993,951,014,270,933,512,030,194,957,221,371,956,828,843,521,568,082,173,786,251,242,333,157,830,450,435,623,211,664,308,500,316,844,478,617,809,101,158,220,672,108,895,053,508,829,266,120,497,031,742,749,376,045,929,890,296,052,805,527,212,315,382,805,219,353,316,270,742,572,401,962,035,464,878,235,703,759,464,796,806,075,131,056,520,079,836,955,770,415,021,318,508,272,982,103,736,658,633,390,411,347,759,000,563,271,226,062,182,345,964,184,167,346,918,225,243,856,348,794,013,355,418,404,695,826,256,911,622,054,015,423,611,375,261,945,905,974,225,257,659,010,379,414,787,547,681,984,112,941,581,325,198,396,634,685,659,217,861,208,771,400,322,507,388,161,967,513,719,166,366,839,894,214,040,787,733,471,287,845,629,833,993,885,413,462,225,294,548,785,581,641,804,620,417,256,563,685,280,586,511,301,918,399,010,451,347,815,776,570,842,790,738,545,306,707,750,937,624,267,501,103,840,324,470,083,425,714,138,183,905,657,667,736,579,430,274,197,734,179,172,691,637,931,540,695,631,396,056,193,786,415,805,463,680,000

XPL0

int N, D;
[N:= 2*3*5*7*11*13*17*19;
D:= 1;
repeat  D:= D+1;
        if rem(N/D) then
            [D:= 1;  N:= N + 2*3*5*7*11*13*17*19];
until   D = 20;
IntOut(0, N);
]

Output:

232792560

Yabasic

Translation of: XPL0

// Rosetta Code problem: http://rosettacode.org/wiki/Smallest_multiple
// by Galileo, 05/2022

M = 2*3*5*7*11*13*17*19
N = M
D = 1
repeat
    D = D + 1
    if mod(N, D) D = 1 : N = N + M
until D = 20
print N

Output:

232792560
---Program done, press RETURN---