How can we calculate 2¹⁰⁰(2^100)?

By Mr. Subramani K, Project Associate @IIITB

Standard method is to multiplying 2, by 100 times that results:

1267650600228229401496703205376

Now, we have an interesting method of obtaining 2^100 without multiplication, which is explained as:

Step 1: 2^10 (2¹⁰)

• Write digit “1”, and add suffix “000”, then it becomes 1000
• Dividing the number by 50, 1000/50 =20
• Again, dividing by 5, 20/5 =4
• Then add all results: 1000 + 20 +4 =1024= 2^10 (2¹⁰)
• Begin the next step with the result of this step.

Step 2: 2^20 (2²⁰)

• Write digit “1024”, and add suffix “000”, then it becomes 1024000
• Dividing the number by 50, 1024000/50 =20480
• Again, dividing by 5, 20480/5 =4096
• Then add all results: 1024000 + 20480 +4096 =1048576= 2^20 (2²⁰)

Similarly,

Step 3: 2^30 = 1073741824

Step 4: 2^40 = 1099511627776

Step 5: 2^50 = 1125899906842624

Step 6: 2^60 = 1152921504606846976

Step 7: 2^70 = 1180591620717411303424

Step 8: 2^80 = 1208925819614629174706176

Step 9: 2^90 = 1237940039285380274899124224

                          Begin the next step with the result of this step using the same logic.

Step 10: 2^100 =

1237940039285380274899124224000

 24758800785707605497982484480

4951760157141521099596496896

——————————————————-

1267650600228229401496703205376 = 2 ^ 100

We can conclude that the operation 2¹⁰⁰(2^100) can be performed without using any multiplication and only with the help of few additions and divisions.