Wednesday, August 22

LCM


Cliff Pickover's tweet from 2015 examples a least common multiple (LCM) of the positive integers to 100. Folk of my generation first learned of LCM in school when adding fractions with differing denominators. Finding the LCM of a range of numbers is fast and easy in Mathematica:

LCM @@ Range[100]
69720375229712477164533808935312303556800

One could look up the answer in OEIS sequence A003418 (its current b-file goes up to 2308). But of course I wanted to know how much further I could take this:

LCM @@ Range[10000]
57933396702876429686922708791662400986348602979985188253931383511489793001457731823088325981761829221665744176794023407056559491402467891577328326763021299466843118474637852656831938521549472347971073068161679301705472685236926463387338495220571064420250677315000599457941340849496227227628926493771018264821842230370349640102573492881424317306189569467101495834601991270039918780924506495405797923762205360790652073159333382795670426041033566699342449050309786673681670483369155689567554239898879039744147333971988258061042090970476729293484513072443614795766878726325795854855394491290821167148355514749149683707585283381546153703014210442470318180511906691108325146494219343498899382918018246586609827667470329166012110874981104800415741527586280026737848182673635645872230905234515169611121042867043956727839314198728626274066655467846183343599194761590368608472578398169740111485924046986870714883894285841394964627408094161019230662749101230783008668676907211199488107523306410531772045452853957706873238466829988649822157557103503283563398281775464911904789159515900987401574678885942493907604740891878907698622679570965569483682456042918236444719794534411171907606336090534029349351300276141892529795448751826394399153216183270385737795748770508612096374765333578237973395907265484337502903901947799663388329849198045756207969590055686607678195206367273600632909417024224754750428711236917913663419215925830944035539848749163178489614227546656090790164108195741048033614368495827231281392190063051315248070192263400801315095608512139510731469732311313898995746040563433121427776071482655904346538281010668476731132415829844984600414136781404774213539507859790229205890271721600309169926806121871750008163738773911610009508609149665332579632767397078877996926581337419351834754370411008686136818501030862345505385357198060894463821342298717851567836562984344806469613768024764967372979655179066074398198246805104576134474823016488842818077041661676098399378809713894284994865370648616800689225595431967181072865363430005250840767890912164530705704936837915584856606960687347372391339254432119085932175541392954343684716695162629271229789289404752104218596977036941910521266321726821940533986384237994403780618301379099347975260122724194454275088825587044488208965690373706904056926509324696308810974331790119456438147168585552011926921912167450509941646104076818762060881903969616431646384985895944231218505620547093874241169759205450145478746112796898626711966320965057212219958567338851356631739947125096250452942497473309299907612330435197454392788637359253116308685007014249605492659524429134513344137517101872279428202285951652856354827230765931502805341696470148698002737700823078904634554776750169178259216255903968865588749827789888950172452455448248833712309835657561369233157977405579365293671943131412034109901944892819245001657496671822581274180596255340507054499934060282320458240722454209933569735940032859109934686878274110864394924463573852015338428881961843292083566034669814619612606638283615766521897504566616272305253931938372830446073384019299355320864342734019517633662346790422915951954822645137091494126100390104510987373366328615363056042137440808225973600809566845718073791616927784260557845021823094999326904373592319407516660896764388092262510369182153559285446074990941863516247226532653142198551840063631989428776799533286215464660644129411503287306838551341184739976807097763115368031748646043780549055143428297230678053738453010234949008253769355207208167999203353157524666017029803679612131824740794652592875662818479980117505768541194835524231818203552256426752730455115752280837099763237606348192867936457993970866446264015812819179994138642295108872381709181937092290392544335464025324661284746003660247161196698209062164637264114930766444473471083408200329662059064201896721165015687487728300854501780810155844837489798144309942999091774466406270065305461848242329380636274754660519867343112275861821293501112101434868225378041813836808745417606289159904294165941408692922250601127804971962342807927743390030395048263275616935165347620718001157478088456439083590834464409622781693790883289597024043982584220069224170235863458745344365684082114430362867446193601075569803650773018026700003812298460527976219100308016537538008597751565631582745643139434508332515569645426771932483266712323523039014220800000

LCM @@ Range[1000000000]
abridged answer

If you want the unabridged answer to the LCM of the positive integers to one billion, it's 440 MB — somewhat unsuitable for web-browser display. If you have a text application that can handle that, I've put a 207 MB .zip compression of it here [clicking on this should download the compressed file]. I discovered an interesting initial-digits convergence when I calculated the number of decimal digits in these to-powers-of-ten LCMs:

Table[Ceiling[Log[10, LCM @@ Range[10^n]]], {n, 9}]
{4,41,433,4349,43452,434115,4342311,43428686,434295176}

The initial 4 is the number of digits in 2520 (LCM to 10) and the next 41 is the number of digits in LCM to 100. The final 434295176 is the number of digits in LCM to one billion.

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