[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-359305-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-359305-en":131},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","least-common-multiple-word-problems-a-solve-word-problems","Least Common Multiple Word Problems (A) - Solve Word Problems","","A set of eight least common multiple word problems set in music contexts, asking when different rhythms or schedules will coincide again. Learners solve timing questions involving drum hits, stage performances, a piano chord and bell, rotating performers, overlapping spotlight colors, lesson days, recording schedules for two bands, and rotating concert events across days. An answer key provides final times/days and the LCM reasoning steps for each problem.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/exam/","Exam",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/least-common-multiple-word-problems-a-solve-word-problems/359305/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/least-common-multiple-word-problems-a-solve-word-problems/359305.png","ImageObject",300,407,{"name":42,"@type":43},"Jacob","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-11-02","2026-09-23",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",15,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"How do you find when two events happen again at the same time?","Question",{"text":63,"@type":64},"Compute the LCM of the two intervals (beats, minutes, seconds, or days). The LCM gives the next time both events coincide.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"Why does the answer key use prime factorization in some problems?",{"text":68,"@type":64},"Prime factorization helps calculate the LCM accurately by combining the highest powers of each prime factor from the given intervals.",{"name":70,"@type":61,"acceptedAnswer":71},"In problem 3, how many times does the chord play within the shared time?",{"text":72,"@type":64},"The LCM of 8 seconds and 12 seconds is 24 seconds. The chord plays every 8 seconds, so it occurs at 8, 16, and 24—3 times.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},359305,1791098712,{"code":4,"msg":82,"data":83},"success",[84,88,92,95,100,105,110,115,120,123,127],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":93,"slug":94},70,"exam",{"id":96,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},5,"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},6,"Technology",50,"technology",{"id":106,"doc_module":4,"doc_module_name":25,"category_name":107,"show_sort_weight":108,"slug":109},7,"Healthcare",40,"healthcare",{"id":111,"doc_module":4,"doc_module_name":25,"category_name":112,"show_sort_weight":113,"slug":114},8,"Research & Report",30,"research-report",{"id":116,"doc_module":4,"doc_module_name":25,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":25,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":25,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":25,"category_name":129,"show_sort_weight":96,"slug":130},19,"General","general",{"code":4,"msg":82,"data":132},{"doc_id":79,"user_id":133,"nickname":42,"user_avatar":134,"doc_module":4,"category_id":33,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":33,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":144,"read_time":124,"preview_page_count":4},962084931830,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","Name:    \nLeast Common Multiple Word Problems (A)  \nMUSIC & PERFORMANCES  \nDirections: Solve each word problem. Show all of your work and circle your answer.  \n1. A drummer hits a bass drum every 4 beats and a snare drum every 6 beats.  \nIf both drums are hit at beat 1, after how many beats will they both be hit at the same time again?  \nShow your work:  \n2. At a music festival, Stage A has a new performance every 15 minutes and Stage B has a new performance every 20 minutes. If both stages start at the same time, how many minutes until they both start a new performance simultaneously?  \nShow your work:  \n3. A music teacher plays a chord on the piano every 8 seconds and rings a bell every 12 seconds. Starting at the same moment, how many seconds until both happen at the same time again? How many times will she play the chord in that time?  \nShow  \nyour work:  \nMashup Math LLC | Al[l Rights Reserved | Get More K-12 Math Worksheets at www.mashupmath.com](l Rights Reserved | Get More K-12 Math Worksheets at www.mashupmath.com)  \n4. Three performers rotate on stage: a singer every 6 minutes, a dancer every 9 minutes, and a magician every 4 minutes. If all three start at the same time, after how many minutes will all three perform at the same time again?  \nShow your work:  \n5. A spotlight cycles red every 10 seconds, blue every 15 seconds, and green every 6 seconds. If all three flash at time zero, after how many seconds will all three flash together again?  \nShow your work:  \n6. A music school holds guitar lessons every 8 days and piano lessons every 12 days. Both are held on Day 1. On what day will both next be held on the same day? How many of each lesson will have  \noccurred by then?  \nShow your work:  \nMashup Math LLC | Al[l Rights Reserved | Get More K-12 Math Worksheets at www.mashupmath.com](l Rights Reserved | Get More K-12 Math Worksheets at www.mashupmath.com)  \n7. Two bands are recording. Band A records a new song every 18 days and Band B every 24 days. Both record on Day 1. What is the next day both will record on the same day? How many songs will  \neach band have recorded by then (including Day 1)?  \nShow your work:  \nChallenge Problem  \n8. A concert hall hosts three events on rotation: jazz concerts every 14 days, classical concerts every  \n21 days, and pop concerts every 35 days. All three are held on Day 1. What is the next day all three will be held on the same day? How many of each event will have occurred by then (including Day 1)?  \nShow your work:  \nMashup Math LLC | Al[l Rights Reserved | Get More K-12 Math Worksheets at www.mashupmath.com](l Rights Reserved | Get More K-12 Math Worksheets at www.mashupmath.com)  \nANSWER KEY  \n1.) 1. ) Answer: Beat 12 Steps: Multiples of 4: 4, 8, 12, ... → Multiples of 6: 6, 12, ... → LCM(4,6) = 12  \n2.) 2. ) Answer: 60 minutes Steps: 15 = 3×5 20 = 2² ×5 → LCM(15,20) = 2² ×3×5 = 60 minutes  \n3.) 3. ) Answer: 24 seconds; chord plays 3 times Steps: 8 = 2³ 12 = 2² ×3 → LCM(8,12) = 2³ ×3 = 24 seconds → Chord plays at 8, 16, 24 → 3 times in 24 seconds  \n4.) 4. ) Answer: 36 minutes Steps: LCM(6,9): 6=2×3, 9=3² → LCM=2×3²=18 → LCM(18,4): 4=2² → LCM=2² ×3²=36 minutes  \n5.) 5. ) Answer: 30 seconds Steps: LCM(10,15): 10=2×5, 15=3×5 → LCM=2×3×5=30 → LCM(30,6):  \n6=2×3 → LCM=30 seconds  \n6.) 6. ) Answer: Day 24; guitar 3 times, piano 2 times Steps: LCM(8,12) = 2³ ×3 = 24 → Both meet again on Day 24 → Guitar: 24÷8 = 3 lessons Piano: 24÷12 = 2 lessons  \n7.) 7. ) Answer: Day 72; Band A: 4 songs, Band B: 3 songs Steps: 18 = 2×3² 24 = 2³ ×3 → LCM(18,24) = 2³ ×3² = 72 → Next shared day: Day 72 → Band A: 72÷18 = 4 songs Band B: 72÷24 = 3 songs  \n8.) 8. ) Answer: Day 210; Jazz: 15, Classical: 10, Pop: 6 Steps: LCM(14,21): 14=2×7, 21=3×7 → LCM=2×3×7=42 → LCM(42,35): 35=5×7 → LCM=2×3×5×7=210 → Jazz: 210÷14=15 Classical: 210÷21=10 Pop: 210÷35=6  \nMashup Math LLC | Al[l Rights Reserved | Get More K-12 Math Worksheets at www.mashupmath.com](l Rights Reserved | Get More K-12 Math Worksheets at www.mas","cbCaitqvBMrc6Zqb","https://ap.wps.com/l/cbCaitqvBMrc6Zqb","pdf",1342191,"English","# Directions\n## Problems 1-8\n# Answer Key","[{\"question\":\"How do you find when two events happen again at the same time?\",\"answer\":\"Compute the LCM of the two intervals (beats, minutes, seconds, or days). The LCM gives the next time both events coincide.\"},{\"question\":\"Why does the answer key use prime factorization in some problems?\",\"answer\":\"Prime factorization helps calculate the LCM accurately by combining the highest powers of each prime factor from the given intervals.\"},{\"question\":\"In problem 3, how many times does the chord play within the shared time?\",\"answer\":\"The LCM of 8 seconds and 12 seconds is 24 seconds. The chord plays every 8 seconds, so it occurs at 8, 16, and 24—3 times.\"}]","Least Common Multiple Word Problems (A) - Solve Word Problems | PDF",1790136657]