Solve (83!/81!+82!)(40!+41!/42!

$$( \frac{ 83 ! }{ 81 ! +82 ! } )( \frac{ 40 ! +41 ! }{ 42 ! }$$

Evaluate

$$2$$

Solution Steps

The factorial of 83 is 39455239697206586511897471180120610571436503407643446275224357528369751562996629334879591940103770870906880000000000000000000.

$$\frac{39455239697206586511897471180120610571436503407643446275224357528369751562996629334879591940103770870906880000000000000000000}{81!+82!}\times \frac{40!+41!}{42!}$$

The factorial of 81 is 5797126020747367985879734231578109105412357244731625958745865049716390179693892056256184534249745940480000000000000000000.

$$\frac{39455239697206586511897471180120610571436503407643446275224357528369751562996629334879591940103770870906880000000000000000000}{5797126020747367985879734231578109105412357244731625958745865049716390179693892056256184534249745940480000000000000000000+82!}\times \frac{40!+41!}{42!}$$

The factorial of 82 is 475364333701284174842138206989404946643813294067993328617160934076743994734899148613007131808479167119360000000000000000000.

$$\frac{39455239697206586511897471180120610571436503407643446275224357528369751562996629334879591940103770870906880000000000000000000}{5797126020747367985879734231578109105412357244731625958745865049716390179693892056256184534249745940480000000000000000000+475364333701284174842138206989404946643813294067993328617160934076743994734899148613007131808479167119360000000000000000000}\times \frac{40!+41!}{42!}$$

Add 5797126020747367985879734231578109105412357244731625958745865049716390179693892056256184534249745940480000000000000000000 and 475364333701284174842138206989404946643813294067993328617160934076743994734899148613007131808479167119360000000000000000000 to get 481161459722031542828017941220983055749225651312724954575906799126460384914593040669263316342728913059840000000000000000000.

$$\frac{39455239697206586511897471180120610571436503407643446275224357528369751562996629334879591940103770870906880000000000000000000}{481161459722031542828017941220983055749225651312724954575906799126460384914593040669263316342728913059840000000000000000000}\times \frac{40!+41!}{42!}$$

Divide 39455239697206586511897471180120610571436503407643446275224357528369751562996629334879591940103770870906880000000000000000000 by 481161459722031542828017941220983055749225651312724954575906799126460384914593040669263316342728913059840000000000000000000 to get 82.

$$82\times \frac{40!+41!}{42!}$$

The factorial of 40 is 815915283247897734345611269596115894272000000000.

$$82\times \frac{815915283247897734345611269596115894272000000000+41!}{42!}$$

The factorial of 41 is 33452526613163807108170062053440751665152000000000.

$$82\times \frac{815915283247897734345611269596115894272000000000+33452526613163807108170062053440751665152000000000}{42!}$$

Add 815915283247897734345611269596115894272000000000 and 33452526613163807108170062053440751665152000000000 to get 34268441896411704842515673323036867559424000000000.

$$82\times \frac{34268441896411704842515673323036867559424000000000}{42!}$$

The factorial of 42 is 1405006117752879898543142606244511569936384000000000.

$$82\times \frac{34268441896411704842515673323036867559424000000000}{1405006117752879898543142606244511569936384000000000}$$

Reduce the fraction \frac{34268441896411704842515673323036867559424000000000}{1405006117752879898543142606244511569936384000000000} to lowest terms by extracting and canceling out 34268441896411704842515673323036867559424000000000.

$$82\times \frac{1}{41}$$

Multiply 82 and \frac{1}{41} to get \frac{82}{41}.

$$\frac{82}{41}$$

Divide 82 by 41 to get 2.

$$2$$

Factor

$$2$$