Determine the smallest positive integer which is 2 times a square, 3 times a cube, 5 times a fifth power, 7 times a seventh power, and 11 times an eleventh power.
For example, 648 is 2 times the square of 18, and 3 times the cube of 6.
(In reply to
solution by Charlie)
For a number to be p times a pth power, the exponent of p must be 1 more than a multiple of p. 1155 is 1 more than a multiple of 2, 330 is 1 more than a multiple of 7, and 210 is 1 more than a multiple of 11, but the smallest multiple of 770 that is 1 more than a multiple of 3 is 1540, and the smallest multiple of 462 that is 1 more than a multiple of 5 is 1386. The correct answer is 2^1155*3^1540*5^1386*7^330*11^210.
2*(2^577*3^770*5^693*7^165*11^105)^2=2*2^1154*2^1540*5^1386*7^330*11^210=2^1155*3^1540*5^1386*7^330*11^210
3*(2^385*3^513*5^462*7^110*11^70)^3=3*2^1155*3^1539*5^1386*7^330*11^210=2^1155*3^1540*5^1386*7^330*11^210
5*(2^231*3^308*5^277*7^66*11^42)^5=5*2^1155*3^1540*5^1385*7^330*11^210=2^1155*3^1540*5^1386*7^330*11^210
7*(2^165*3^220*5^198*7^47*11^30)^7=7*2*1155*3^1540*5^1386*7^329*11^210=2^1155*3^1540*5^1386*7^330*11^210
11*(2^105*3^140*5^126*7^30*11^19)^11=11*2^1155*3^1540*5^1386*7^330*11^209=2^1155*3^1540*5^1386*7^330*11^210
2^1155*3^1540*5^1386*7^330*11^210=63623910399076754772410007909772874292974508545924666862999049646003149311211898721368702402675193492362236607572351184159847467297678563454217264705067313227121415260318420891719346984155848231239825804870168979383166636240081645070828008288734748473561479477898454495334660942977835098826614720029773429440419374314114504625678221235453056270090231996308849413718272287382291974254808217747504405573280533409945313642414691893528844104761944401626167521232682829461495277095001413371055861874982982274833496826011625018490185548084497611136120945265919325996390215686653604962161816152844368066467623996487875027456860805557133140671952816447532963099528000552074944792383154112322231871291264545276883141517671186236225704001720550140029767280930005031806816893460448560337918028366716448051285703942649567159655526610191192689307422771208858696144678609395116780511349905462119393871703246958095168982553691354441229032421000493768093456333138390891150907242773595952266336901729153417868226061382309097688627098513878410033141302871408678813586953646658522277546989965135765976821737839802715951402292265821048074127805388358348936576286608756497805773109058632015920748240713801530049740111346384111002612714138997677918936077965051907500484673016816283440702531475562751308892388205040516936642994559716496776094362874270748301793024507987640782553928620046690411982126533985137939453125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
By the way, the number in your post appears to be 2^1156*3^771*5^463*7*11^211. Here is the correct value of 2^1155*3^770*5^462*7^330*11^210.
2^1155*3^770*5^462*7^330*11^210=373224562145818877146946454967221083604147764461322624157524620720085456807967560982473494607918057574631001646694871454753271521307752008227916825543989657066094944334362939801708358392520023168024511426867324435770955736620806147599458047497113966241281835393986421129210906276390439926201714496741971939667018150014794404731333951670659428410190432718515200664159335518361514944633053591937749795416682159145834595572666755373718770763335910554737336755910099785410101591817643168264026267111947267741807932263898094276358173149879031139664182588318983514843606169796311269509532515088113704544849164346895892899728944978087124687206476722776493590233215000594423836670016659635016492139126065357643548704508772680441253091539164278028938980937443355843093470049258192488822275620367556420031881224156531215166755009736940334085982515034941629980274761658382339026369455379607134911279272175015926683366607374886463945600152393978865534905017973804110489823921412595090657472277436861585760694067265512667939407289555132371264071391170505365115257210808085760691291553792000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
Edited on October 12, 2023, 4:08 pm
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Posted by Math Man
on 2023-10-12 13:53:33 |
re: solution
