How many digits are there in 2^1000 (2 to the power of 1000)?

this is how i worked out the problem

i started with counting number of digits present in the powers of 2.

number of digits present in the first three powers of 2 consist of single digits.next three powers consist of 2 digits each

i.e
2 raised to the power 1,2,3-->1 digit result
2 raised to the power 4,5,6-->2 digit result
2 raised to the power 7,8,9-->3 digit result

2 raised to the power 10,11,12,13-->4 digit result

2 raised to the power 14,15,16-->5 digit result
2 raised to the power 17,18,19-->6 digit result

2 raised to the power 20,21,22,23-->7 digit result

2 raised to the power 24,25,26-->8 digit result
2 raised to the power 27,28,29-->9 digit result

2 raised to the power 30,31,32,33-->10 digit result

do we see a pattern here?
yup,three continuos powers of 2 have same number of digits,but 4 powers of 2(i.e 10,11,12,13 for eg) give 4 digit result.this pattern of 4 powers of two giving the same result repeats after every 3 digit pattern.
i.e 4 digits appear 4 times
7 digits appear 4 times
10 digits appear 4 times
13 digits appear 4 times so on and so forth...

since 2^1000 falls under this 4 times repeating pattern,I, just applied a direct proportion here,

for every increase of 10 powers resultant answer increases by 3 digits

for eg.2^10 has 4 digits,thus 2^1000 which is 990 times greater than this power has {(990*3)/10 +4} number of digits which results in 301 digits.

phew iam through!!!!!!!!

Posted by akila on 2003-03-01 17:06:42