Solve 312div6!*(306)!

$$312 \div 6 ! \times (306) !$$

Evaluate

$$103642065100010195035927808278905160835125736166929318068820864919289496796714270582139048171808461380021159933835737647827796486518217185895525659017587664183997078730642391187510415600574798017373105419635629917392605968947369283900279536990673465083292089237343884686385018393214623057079201074967035838974105369262254967458949628776491736678973114247495406717418219985422509061463315436552216832612520142427918255086659634118063026093683418732168804789539993510826364122386849807959136766684204845322114907152654009577659389699914104573556128069399347200000000000000000000000000000000000000000000000000000000000000000000000000$$

Solution Steps

The factorial of 6 is 720.

$$\frac{312}{720}\times 306!$$

Reduce the fraction \frac{312}{720} to lowest terms by extracting and canceling out 24.

$$\frac{13}{30}\times 306!$$

The factorial of 306 is 239173996384638911621371865259011909619520929615990734004971226736821915684725239804936265011865680107741138308851702264217991891965116582835828443886740763501531720147636287355793266770557226193937935583774530578598321466801621424385260469978477227115289436701562810814734657830495283977875079403770082705324858544451357617212960681791904007720707186724989400117118969197128867064915343315120500382951969559449042127123060694118606983293115581689620318745092292717291609513200422633751854076963549643051034401121509252871521668538263318246667987852460032000000000000000000000000000000000000000000000000000000000000000000000000000.

$$\frac{13}{30}\times 239173996384638911621371865259011909619520929615990734004971226736821915684725239804936265011865680107741138308851702264217991891965116582835828443886740763501531720147636287355793266770557226193937935583774530578598321466801621424385260469978477227115289436701562810814734657830495283977875079403770082705324858544451357617212960681791904007720707186724989400117118969197128867064915343315120500382951969559449042127123060694118606983293115581689620318745092292717291609513200422633751854076963549643051034401121509252871521668538263318246667987852460032000000000000000000000000000000000000000000000000000000000000000000000000000$$

Express \frac{13}{30}\times 239173996384638911621371865259011909619520929615990734004971226736821915684725239804936265011865680107741138308851702264217991891965116582835828443886740763501531720147636287355793266770557226193937935583774530578598321466801621424385260469978477227115289436701562810814734657830495283977875079403770082705324858544451357617212960681791904007720707186724989400117118969197128867064915343315120500382951969559449042127123060694118606983293115581689620318745092292717291609513200422633751854076963549643051034401121509252871521668538263318246667987852460032000000000000000000000000000000000000000000000000000000000000000000000000000 as a single fraction.

$$\frac{13\times 239173996384638911621371865259011909619520929615990734004971226736821915684725239804936265011865680107741138308851702264217991891965116582835828443886740763501531720147636287355793266770557226193937935583774530578598321466801621424385260469978477227115289436701562810814734657830495283977875079403770082705324858544451357617212960681791904007720707186724989400117118969197128867064915343315120500382951969559449042127123060694118606983293115581689620318745092292717291609513200422633751854076963549643051034401121509252871521668538263318246667987852460032000000000000000000000000000000000000000000000000000000000000000000000000000}{30}$$

Multiply 13 and 239173996384638911621371865259011909619520929615990734004971226736821915684725239804936265011865680107741138308851702264217991891965116582835828443886740763501531720147636287355793266770557226193937935583774530578598321466801621424385260469978477227115289436701562810814734657830495283977875079403770082705324858544451357617212960681791904007720707186724989400117118969197128867064915343315120500382951969559449042127123060694118606983293115581689620318745092292717291609513200422633751854076963549643051034401121509252871521668538263318246667987852460032000000000000000000000000000000000000000000000000000000000000000000000000000 to get 3109261953000305851077834248367154825053772085007879542064625947578684903901428117464171445154253841400634798015072129434833894595546515576865769770527629925519912361919271735625312468017243940521193162589068897521778179068421078517008386109720203952498762677120316540591550551796438691712376032249011075169223161077867649023768488863294752100369193427424862201522546599562675271843899463096566504978375604272837547652599789023541890782810502561965064143686199805324790923671605494238774103000526145359663447214579620287329781690997423137206683842081980416000000000000000000000000000000000000000000000000000000000000000000000000000.

$$\frac{3109261953000305851077834248367154825053772085007879542064625947578684903901428117464171445154253841400634798015072129434833894595546515576865769770527629925519912361919271735625312468017243940521193162589068897521778179068421078517008386109720203952498762677120316540591550551796438691712376032249011075169223161077867649023768488863294752100369193427424862201522546599562675271843899463096566504978375604272837547652599789023541890782810502561965064143686199805324790923671605494238774103000526145359663447214579620287329781690997423137206683842081980416000000000000000000000000000000000000000000000000000000000000000000000000000}{30}$$

Divide 3109261953000305851077834248367154825053772085007879542064625947578684903901428117464171445154253841400634798015072129434833894595546515576865769770527629925519912361919271735625312468017243940521193162589068897521778179068421078517008386109720203952498762677120316540591550551796438691712376032249011075169223161077867649023768488863294752100369193427424862201522546599562675271843899463096566504978375604272837547652599789023541890782810502561965064143686199805324790923671605494238774103000526145359663447214579620287329781690997423137206683842081980416000000000000000000000000000000000000000000000000000000000000000000000000000 by 30 to get 103642065100010195035927808278905160835125736166929318068820864919289496796714270582139048171808461380021159933835737647827796486518217185895525659017587664183997078730642391187510415600574798017373105419635629917392605968947369283900279536990673465083292089237343884686385018393214623057079201074967035838974105369262254967458949628776491736678973114247495406717418219985422509061463315436552216832612520142427918255086659634118063026093683418732168804789539993510826364122386849807959136766684204845322114907152654009577659389699914104573556128069399347200000000000000000000000000000000000000000000000000000000000000000000000000.

$$103642065100010195035927808278905160835125736166929318068820864919289496796714270582139048171808461380021159933835737647827796486518217185895525659017587664183997078730642391187510415600574798017373105419635629917392605968947369283900279536990673465083292089237343884686385018393214623057079201074967035838974105369262254967458949628776491736678973114247495406717418219985422509061463315436552216832612520142427918255086659634118063026093683418732168804789539993510826364122386849807959136766684204845322114907152654009577659389699914104573556128069399347200000000000000000000000000000000000000000000000000000000000000000000000000$$

Factor

$$2^{301}\times 3^{150}\times 5^{74}\times 7^{49}\times 11^{29}\times 13^{25}\times 17^{19}\times 19^{16}\times 23^{13}\times 29^{10}\times 31^{9}\times 37^{8}\times 41^{7}\times 43^{7}\times 47^{6}\times 53^{5}\times 59^{5}\times 61^{5}\times 67^{4}\times 71^{4}\times 73^{4}\times 79^{3}\times 83^{3}\times 89^{3}\times 97^{3}\times 101^{3}\times 103^{2}\times 107^{2}\times 109^{2}\times 113^{2}\times 127^{2}\times 131^{2}\times 137^{2}\times 139^{2}\times 149^{2}\times 151^{2}\times 157\times 163\times 167\times 173\times 179\times 181\times 191\times 193\times 197\times 199\times 211\times 223\times 227\times 229\times 233\times 239\times 241\times 251\times 257\times 263\times 269\times 271\times 277\times 281\times 283\times 293$$